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Integer in the infinite Fibonacci sequence

A tiling with squares whose side lengths are successive Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13 and 21.

In mathematics, the Fibonacci numbers, commonly denoted F_n , form a sequence, called the Fibonacci sequence, such that each number is the sum of the two preceding ones, starting from 0 and 1. That is,^[1]

$${\displaystyle F_{0}=0,\quad F_{1}=1,}$$

and

$${\displaystyle F_{n}=F_{n-1}+F_{n-2}}$$

for n > 1.

The sequence starts:^[2]

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ...

Under some older definitions, the value ${\displaystyle F_{0}=0}$ is omitted, so that the sequence starts with ${\displaystyle F_{1}=F_{2}=1,}$ and the recurrence ${\displaystyle F_{n}=F_{n-1}+F_{n-2}}$ is valid for n > 2.^{[3] [4]} In his original definition, Fibonacci started the sequence with ${\displaystyle F_{1}=1,F_{2}=2}$ ^[5]

The Fibonacci spiral: an approximation of the golden spiral created by drawing circular arcs connecting the opposite corners of squares in the Fibonacci tiling; (see preceding image)

Fibonacci numbers are strongly related to the golden ratio: Binet's formula expresses the nth Fibonacci number in terms of n and the golden ratio, and implies that the ratio of two consecutive Fibonacci numbers tends to the golden ratio as n increases.

Fibonacci numbers are named after the Italian mathematician Leonardo of Pisa, later known as Fibonacci. In his 1202 book Liber Abaci, Fibonacci introduced the sequence to Western European mathematics,^[6] although the sequence had been described earlier in Indian mathematics,^{[7] [8] [9]} as early as 200 BC in work by Pingala on enumerating possible patterns of Sanskrit poetry formed from syllables of two lengths.

Fibonacci numbers appear unexpectedly often in mathematics, so much so that there is an entire journal dedicated to their study, the Fibonacci Quarterly. Applications of Fibonacci numbers include computer algorithms such as the Fibonacci search technique and the Fibonacci heap data structure, and graphs called Fibonacci cubes used for interconnecting parallel and distributed systems.

They also appear in biological settings, such as branching in trees, the arrangement of leaves on a stem, the fruit sprouts of a pineapple, the flowering of an artichoke, an uncurling fern, and the arrangement of a pine cone's bracts.

Fibonacci numbers are also closely related to Lucas numbers ${\displaystyle L_{n}}$ , in that the Fibonacci and Lucas numbers form a complementary pair of Lucas sequences: ${\displaystyle U_{n}(1,-1)=F_{n}}$ and ${\displaystyle V_{n}(1,-1)=L_{n}}$ .

History [edit]

Thirteen (F ₇) ways of arranging long (shown by the red tiles) and short syllables (shown by the grey squares) in a cadence of length six. Five (F ₅) end with a long syllable and eight (F ₆) end with a short syllable.

The Fibonacci sequence appears in Indian mathematics in connection with Sanskrit prosody, as pointed out by Parmanand Singh in 1986.^{[8] [10] [11]} In the Sanskrit poetic tradition, there was interest in enumerating all patterns of long (L) syllables of 2 units duration, juxtaposed with short (S) syllables of 1 unit duration. Counting the different patterns of successive L and S with a given total duration results in the Fibonacci numbers: the number of patterns of duration m units is F _{m + 1} .^[9]

Knowledge of the Fibonacci sequence was expressed as early as Pingala (c. 450 BC–200 BC). Singh cites Pingala's cryptic formula misrau cha ("the two are mixed") and scholars who interpret it in context as saying that the number of patterns for m beats ( F _m+1 ) is obtained by adding one [S] to the F _m cases and one [L] to the F _m−1 cases.^[12] Bharata Muni also expresses knowledge of the sequence in the Natya Shastra (c. 100 BC–c. 350 AD).^{[13] [7]} However, the clearest exposition of the sequence arises in the work of Virahanka (c. 700 AD), whose own work is lost, but is available in a quotation by Gopala (c. 1135):^[11]

Variations of two earlier meters [is the variation]... For example, for [a meter of length] four, variations of meters of two [and] three being mixed, five happens. [works out examples 8, 13, 21]... In this way, the process should be followed in all mātrā-vṛttas [prosodic combinations].^[a]

Hemachandra (c. 1150) is credited with knowledge of the sequence as well,^[7] writing that "the sum of the last and the one before the last is the number ... of the next mātrā-vṛtta."^{[15] [16]}

The number of rabbit pairs form the Fibonacci sequence

Outside India, the Fibonacci sequence first appears in the book Liber Abaci (The Book of Calculation, 1202) by Fibonacci^{[6] [17]} where it is used to calculate the growth of rabbit populations.^{[18] [19]} Fibonacci considers the growth of an idealized (biologically unrealistic) rabbit population, assuming that: a newly born breeding pair of rabbits are put in a field; each breeding pair mates at the age of one month, and at the end of their second month they always produce another pair of rabbits; and rabbits never die, but continue breeding forever. Fibonacci posed the puzzle: how many pairs will there be in one year?

• At the end of the first month, they mate, but there is still only 1 pair.
• At the end of the second month they produce a new pair, so there are 2 pairs in the field.
• At the end of the third month, the original pair produce a second pair, but the second pair only mate without breeding, so there are 3 pairs in all.
• At the end of the fourth month, the original pair has produced yet another new pair, and the pair born two months ago also produces their first pair, making 5 pairs.

At the end of the n th month, the number of pairs of rabbits is equal to the number of mature pairs (that is, the number of pairs in month n – 2) plus the number of pairs alive last month (month n – 1). The number in the n th month is the n th Fibonacci number.^[20]

The name "Fibonacci sequence" was first used by the 19th-century number theorist Édouard Lucas.^[21]

Sequence properties [edit]

The first 21 Fibonacci numbers F_n are:^[2]

F 0	F 1	F 2	F 3	F 4	F 5	F 6	F 7	F 8	F 9	F 10	F 11	F 12	F 13	F 14	F 15	F 16	F 17	F 18	F 19	F 20
0	1	1	2	3	5	8	13	21	34	55	89	144	233	377	610	987	1597	2584	4181	6765

The sequence can also be extended to negative index n using the re-arranged recurrence relation

$${\displaystyle F_{n-2}=F_{n}-F_{n-1},}$$

which yields the sequence of "negafibonacci" numbers^[22] satisfying

$${\displaystyle F_{-n}=(-1)^{n+1}F_{n}.}$$

Thus the bidirectional sequence is

F −8

F −7

F −6

F −5

F −4

F −3

F −2

F −1

F 0

F 1

F 2

F 3

F 4

F 5

F 6

F 7

F 8

−21

13

−8

5

−3

2

−1

1

0

1

1

2

3

5

8

13

21

Relation to the golden ratio [edit]

Closed-form expression [edit]

Like every sequence defined by a linear recurrence with constant coefficients, the Fibonacci numbers have a closed-form expression. It has become known as Binet's formula, named after French mathematician Jacques Philippe Marie Binet, though it was already known by Abraham de Moivre and Daniel Bernoulli:^[23]

$${\displaystyle F_{n}={\frac {\varphi ^{n}-\psi ^{n}}{\varphi -\psi }}={\frac {\varphi ^{n}-\psi ^{n}}{\sqrt {5}}},}$$

where

$${\displaystyle \varphi ={\frac {1+{\sqrt {5}}}{2}}\approx 1.61803\,39887\ldots }$$

is the golden ratio (OEIS: A001622), and^[24]

$${\displaystyle \psi ={\frac {1-{\sqrt {5}}}{2}}=1-\varphi =-{1 \over \varphi }\approx -0.61803\,39887\ldots .}$$

Since ${\displaystyle \psi =-\varphi ^{-1}}$ , this formula can also be written as

$${\displaystyle F_{n}={\frac {\varphi ^{n}-(-\varphi )^{-n}}{\sqrt {5}}}={\frac {\varphi ^{n}-(-\varphi )^{-n}}{2\varphi -1}}.}$$

To see this,^[25] note that φ and ψ are both solutions of the equations

$${\displaystyle x^{2}=x+1\quad {\text{and}}\quad x^{n}=x^{n-1}+x^{n-2},}$$

so the powers of φ and ψ satisfy the Fibonacci recursion. In other words,

$${\displaystyle \varphi ^{n}=\varphi ^{n-1}+\varphi ^{n-2}}$$

and

$${\displaystyle \psi ^{n}=\psi ^{n-1}+\psi ^{n-2}.}$$

It follows that for any values a and b , the sequence defined by

$${\displaystyle U_{n}=a\varphi ^{n}+b\psi ^{n}}$$

satisfies the same recurrence.

$${\displaystyle U_{n-1}+U_{n-2}=a\varphi ^{n-1}+b\psi ^{n-1}+a\varphi ^{n-2}+b\psi ^{n-2}=a\varphi ^{n-1}+a\varphi ^{n-2}+b\psi ^{n-1}+b\psi ^{n-2}=U_{n}}$$

If a and b are chosen so that U ₀ = 0 and U ₁ = 1 then the resulting sequence U _n must be the Fibonacci sequence. This is the same as requiring a and b satisfy the system of equations:

$${\displaystyle \left\{{\begin{array}{l}a+b=0\\\varphi a+\psi b=1\end{array}}\right.}$$

which has solution

$${\displaystyle a={\frac {1}{\varphi -\psi }}={\frac {1}{\sqrt {5}}},\quad b=-a,}$$

producing the required formula.

Taking the starting values U ₀ and U ₁ to be arbitrary constants, a more general solution is:

$${\displaystyle U_{n}=a\varphi ^{n}+b\psi ^{n}}$$

where

$${\displaystyle a={\frac {U_{1}-U_{0}\psi }{\sqrt {5}}}}$$

$${\displaystyle b={\frac {U_{0}\varphi -U_{1}}{\sqrt {5}}}.}$$

Computation by rounding [edit]

Since

$${\displaystyle \left|{\frac {\psi ^{n}}{\sqrt {5}}}\right|<{\frac {1}{2}}}$$

for all n ≥ 0, the number F _n is the closest integer to ${\displaystyle {\frac {\varphi ^{n}}{\sqrt {5}}}}$ . Therefore, it can be found by rounding, using the nearest integer function:

$${\displaystyle F_{n}=\left[{\frac {\varphi ^{n}}{\sqrt {5}}}\right],\ n\geq 0.}$$

In fact, the rounding error is very small, being less than 0.1 for n ≥ 4, and less than 0.01 for n ≥ 8.

Fibonacci numbers can also be computed by truncation, in terms of the floor function:

$${\displaystyle F_{n}=\left\lfloor {\frac {\varphi ^{n}}{\sqrt {5}}}+{\frac {1}{2}}\right\rfloor ,\ n\geq 0.}$$

As the floor function is monotonic, the latter formula can be inverted for finding the index n(F) of the largest Fibonacci number that is not greater than a real number F > 1:

$${\displaystyle n(F)=\left\lfloor \log _{\varphi }\left(F\cdot {\sqrt {5}}+{\frac {1}{2}}\right)\right\rfloor ,}$$

where ${\displaystyle \log _{\varphi }(x)=\ln(x)/\ln(\varphi )=\log _{10}(x)/\log _{10}(\varphi ).}$

Limit of consecutive quotients [edit]

Johannes Kepler observed that the ratio of consecutive Fibonacci numbers converges. He wrote that "as 5 is to 8 so is 8 to 13, practically, and as 8 is to 13, so is 13 to 21 almost", and concluded that these ratios approach the golden ratio ${\displaystyle \varphi \colon }$ ^{[26] [27]}

$${\displaystyle \lim _{n\to \infty }{\frac {F_{n+1}}{F_{n}}}=\varphi .}$$

This convergence holds regardless of the starting values, excluding 0 and 0, or any pair in the conjugate golden ratio, ${\displaystyle -1/\varphi .}$ ^{[ clarification needed ]} This can be verified using Binet's formula. For example, the initial values 3 and 2 generate the sequence 3, 2, 5, 7, 12, 19, 31, 50, 81, 131, 212, 343, 555, ... The ratio of consecutive terms in this sequence shows the same convergence towards the golden ratio.

Successive tilings of the plane and a graph of approximations to the golden ratio calculated by dividing each Fibonacci number by the previous

Decomposition of powers [edit]

Since the golden ratio satisfies the equation

$${\displaystyle \varphi ^{2}=\varphi +1,}$$

this expression can be used to decompose higher powers ${\displaystyle \varphi ^{n}}$ as a linear function of lower powers, which in turn can be decomposed all the way down to a linear combination of ${\displaystyle \varphi }$ and 1. The resulting recurrence relationships yield Fibonacci numbers as the linear coefficients:

$${\displaystyle \varphi ^{n}=F_{n}\varphi +F_{n-1}.}$$

This equation can be proved by induction on n.

This expression is also true for n < 1 if the Fibonacci sequence F_n is extended to negative integers using the Fibonacci rule ${\displaystyle F_{n}=F_{n-1}+F_{n-2}.}$

Matrix form [edit]

A 2-dimensional system of linear difference equations that describes the Fibonacci sequence is

$${\displaystyle {F_{k+2} \choose F_{k+1}}={\begin{pmatrix}1&1\\1&0\end{pmatrix}}{F_{k+1} \choose F_{k}}}$$

alternatively denoted

$${\displaystyle {\vec {F}}_{k+1}=\mathbf {A} {\vec {F}}_{k},}$$

which yields ${\displaystyle {\vec {F}}_{n}=\mathbf {A} ^{n}{\vec {F}}_{0}}$ . The eigenvalues of the matrix A are ${\displaystyle \varphi ={\frac {1}{2}}(1+{\sqrt {5}})}$ and ${\displaystyle -\varphi ^{-1}={\frac {1}{2}}(1-{\sqrt {5}})}$ corresponding to the respective eigenvectors

$${\displaystyle {\vec {\mu }}={\varphi \choose 1}}$$

and

$${\displaystyle {\vec {\nu }}={-\varphi ^{-1} \choose 1}.}$$

As the initial value is

$${\displaystyle {\vec {F}}_{0}={1 \choose 0}={\frac {1}{\sqrt {5}}}{\vec {\mu }}-{\frac {1}{\sqrt {5}}}{\vec {\nu }},}$$

it follows that the nth term is

$${\displaystyle {\begin{aligned}{\vec {F}}_{n}&={\frac {1}{\sqrt {5}}}A^{n}{\vec {\mu }}-{\frac {1}{\sqrt {5}}}A^{n}{\vec {\nu }}\\&={\frac {1}{\sqrt {5}}}\varphi ^{n}{\vec {\mu }}-{\frac {1}{\sqrt {5}}}(-\varphi )^{-n}{\vec {\nu }}~\\&={\cfrac {1}{\sqrt {5}}}\left({\cfrac {1+{\sqrt {5}}}{2}}\right)^{n}{\varphi \choose 1}-{\cfrac {1}{\sqrt {5}}}\left({\cfrac {1-{\sqrt {5}}}{2}}\right)^{n}{-\varphi ^{-1} \choose 1},\end{aligned}}}$$

From this, the nth element in the Fibonacci series may be read off directly as a closed-form expression:

$${\displaystyle F_{n}={\cfrac {1}{\sqrt {5}}}\left({\cfrac {1+{\sqrt {5}}}{2}}\right)^{n}-{\cfrac {1}{\sqrt {5}}}\left({\cfrac {1-{\sqrt {5}}}{2}}\right)^{n}.}$$

Equivalently, the same computation may performed by diagonalization of A through use of its eigendecomposition:

$${\displaystyle {\begin{aligned}A&=S\Lambda S^{-1},\\A^{n}&=S\Lambda ^{n}S^{-1},\end{aligned}}}$$

where ${\displaystyle \Lambda ={\begin{pmatrix}\varphi &0\\0&-\varphi ^{-1}\end{pmatrix}}}$ and ${\displaystyle S={\begin{pmatrix}\varphi &-\varphi ^{-1}\\1&1\end{pmatrix}}.}$ The closed-form expression for the nth element in the Fibonacci series is therefore given by

$${\displaystyle {\begin{aligned}{F_{n+1} \choose F_{n}}&=A^{n}{F_{1} \choose F_{0}}\\&=S\Lambda ^{n}S^{-1}{F_{1} \choose F_{0}}\\&=S{\begin{pmatrix}\varphi ^{n}&0\\0&(-\varphi )^{-n}\end{pmatrix}}S^{-1}{F_{1} \choose F_{0}}\\&={\begin{pmatrix}\varphi &-\varphi ^{-1}\\1&1\end{pmatrix}}{\begin{pmatrix}\varphi ^{n}&0\\0&(-\varphi )^{-n}\end{pmatrix}}{\frac {1}{\sqrt {5}}}{\begin{pmatrix}1&\varphi ^{-1}\\-1&\varphi \end{pmatrix}}{1 \choose 0},\end{aligned}}}$$

which again yields

$${\displaystyle F_{n}={\cfrac {\varphi ^{n}-(-\varphi )^{-n}}{\sqrt {5}}}.}$$

The matrix A has a determinant of −1, and thus it is a 2×2 unimodular matrix.

This property can be understood in terms of the continued fraction representation for the golden ratio:

$${\displaystyle \varphi =1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+\ddots }}}}}}.}$$

The Fibonacci numbers occur as the ratio of successive convergents of the continued fraction for φ, and the matrix formed from successive convergents of any continued fraction has a determinant of +1 or −1. The matrix representation gives the following closed-form expression for the Fibonacci numbers:

$${\displaystyle {\begin{pmatrix}1&1\\1&0\end{pmatrix}}^{n}={\begin{pmatrix}F_{n+1}&F_{n}\\F_{n}&F_{n-1}\end{pmatrix}}.}$$

Taking the determinant of both sides of this equation yields Cassini's identity,

$${\displaystyle (-1)^{n}=F_{n+1}F_{n-1}-{F_{n}}^{2}.}$$

Moreover, since A ⁿ A ^m = A ^n+m for any square matrix A , the following identities can be derived (they are obtained from two different coefficients of the matrix product, and one may easily deduce the second one from the first one by changing n into n + 1),

$${\displaystyle {\begin{aligned}{F_{m}}{F_{n}}+{F_{m-1}}{F_{n-1}}&=F_{m+n-1},\\F_{m}F_{n+1}+F_{m-1}F_{n}&=F_{m+n}.\end{aligned}}}$$

In particular, with m = n ,

$${\displaystyle {\begin{array}{ll}F_{2n-1}&={F_{n}}^{2}+{F_{n-1}}^{2}\\F_{2n}&=(F_{n-1}+F_{n+1})F_{n}\\&=(2F_{n-1}+F_{n})F_{n}\\&=(2F_{n+1}-F_{n})F_{n}.\end{array}}}$$

These last two identities provide a way to compute Fibonacci numbers recursively in O(log(n)) arithmetic operations and in time O(M(n) log(n)), where M(n) is the time for the multiplication of two numbers of n digits. This matches the time for computing the n th Fibonacci number from the closed-form matrix formula, but with fewer redundant steps if one avoids recomputing an already computed Fibonacci number (recursion with memoization).^[28]

Identification [edit]

The question may arise whether a positive integer x is a Fibonacci number. This is true if and only if at least one of ${\displaystyle 5x^{2}+4}$ or ${\displaystyle 5x^{2}-4}$ is a perfect square.^[29] This is because Binet's formula above can be rearranged to give

$${\displaystyle n=\log _{\varphi }\left({\frac {F_{n}{\sqrt {5}}+{\sqrt {5{F_{n}}^{2}\pm 4}}}{2}}\right),}$$

which allows one to find the position in the sequence of a given Fibonacci number.

This formula must return an integer for all n, so the radical expression must be an integer (otherwise the logarithm does not even return a rational number).

Combinatorial identities [edit]

Combinatorial proofs [edit]

Most identities involving Fibonacci numbers can be proved using combinatorial arguments using the fact that ${\displaystyle F_{n}}$ can be interpreted as the number of [possibly empty] sequences of 1s and 2s whose sum is ${\displaystyle n-1}$ . This can be taken as the definition of ${\displaystyle F_{n}}$ with the conventions ${\displaystyle F_{0}=0}$ , meaning no such sequence exists whose sum is −1, and ${\displaystyle F_{1}=1}$ , meaning the empty sequence "adds up" to 0. In the following, ${\displaystyle |{...}|}$ is the cardinality of a set:

${\displaystyle F_{0}=0=|\{\}|}$

${\displaystyle F_{1}=1=|\{\{\}\}|}$

${\displaystyle F_{2}=1=|\{\{1\}\}|}$

${\displaystyle F_{3}=2=|\{\{1,1\},\{2\}\}|}$

${\displaystyle F_{4}=3=|\{\{1,1,1\},\{1,2\},\{2,1\}\}|}$

${\displaystyle F_{5}=5=|\{\{1,1,1,1\},\{1,1,2\},\{1,2,1\},\{2,1,1\},\{2,2\}\}|}$

In this manner the recurrence relation

$${\displaystyle F_{n}=F_{n-1}+F_{n-2}}$$

may be understood by dividing the ${\displaystyle F_{n}}$ sequences into two non-overlapping sets where all sequences either begin with 1 or 2:

$${\displaystyle F_{n}=|\{\{1,...\},\{1,...\},...\}|+|\{\{2,...\},\{2,...\},...\}|}$$

Excluding the first element, the remaining terms in each sequence sum to ${\displaystyle n-2}$ or ${\displaystyle n-3}$ and the cardinality of each set is ${\displaystyle F_{n-1}}$ or ${\displaystyle F_{n-2}}$ giving a total of ${\displaystyle F_{n-1}+F_{n-2}}$ sequences, showing this is equal to ${\displaystyle F_{n}}$ .

In a similar manner it may be shown that the sum of the first Fibonacci numbers up to the nth is equal to the (n + 2)-nd Fibonacci number minus 1.^[30] In symbols:

$${\displaystyle \sum _{i=1}^{n}F_{i}=F_{n+2}-1}$$

This may be seen by dividing all sequences summing to ${\displaystyle n+1}$ based on the location of the first 2. Specifically, each set consists of those sequences that start ${\displaystyle \{2,...\},\{1,2,...\},...,}$ until the last two sets ${\displaystyle \{\{1,1,...,1,2\}\},\{\{1,1,...,1\}\}}$ each with cardinality 1.

Following the same logic as before, by summing the cardinality of each set we see that

${\displaystyle F_{n+2}=F_{n}+F_{n-1}+...+|\{\{1,1,...,1,2\}\}|+|\{\{1,1,...,1\}\}|}$

... where the last two terms have the value ${\displaystyle F_{1}=1}$ . From this it follows that ${\displaystyle \sum _{i=1}^{n}F_{i}=F_{n+2}-1}$ .

A similar argument, grouping the sums by the position of the first 1 rather than the first 2 gives two more identities:

$${\displaystyle \sum _{i=0}^{n-1}F_{2i+1}=F_{2n}}$$

and

$${\displaystyle \sum _{i=1}^{n}F_{2i}=F_{2n+1}-1.}$$

In words, the sum of the first Fibonacci numbers with odd index up to ${\displaystyle F_{2n-1}}$ is the (2n)th Fibonacci number, and the sum of the first Fibonacci numbers with even index up to ${\displaystyle F_{2n}}$ is the (2n + 1)th Fibonacci number minus 1.^[31]

A different trick may be used to prove

$${\displaystyle \sum _{i=1}^{n}{F_{i}}^{2}=F_{n}F_{n+1}}$$

or in words, the sum of the squares of the first Fibonacci numbers up to ${\displaystyle F_{n}}$ is the product of the nth and (n + 1)th Fibonacci numbers. To see this, begin with a Fibonacci rectangle of size ${\displaystyle F_{n}\times F_{n+1}}$ and decompose it into squares of size ${\displaystyle F_{n},F_{n-1},...,F_{1}}$ ; from this the identity follows by comparing areas:

Symbolic method [edit]

The sequence ${\displaystyle (F_{n})_{n\in \mathbb {N} }}$ is also considered using the symbolic method.^[32] More precisely, this sequence corresponds to a specifiable combinatorial class. The specification of this sequence is ${\displaystyle \operatorname {Seq} ({\mathcal {Z+Z^{2}}})}$ . Indeed, as stated above, the ${\displaystyle n}$ -th Fibonacci number equals the number of combinatorial compositions (ordered partitions) of ${\displaystyle n-1}$ using terms 1 and 2.

It follows that the ordinary generating function of the Fibonacci sequence, i.e. ${\displaystyle \sum _{i=0}^{\infty }F_{i}z^{i}}$ , is the complex function ${\displaystyle {\frac {z}{1-z-z^{2}}}}$ .

Induction proofs [edit]

Fibonacci identities often can be easily proved used mathematical induction.

For example, reconsider

$${\displaystyle \sum _{i=1}^{n}F_{i}=F_{n+2}-1.}$$

Adding ${\displaystyle F_{n+1}}$ to both sides gives

${\displaystyle \sum _{i=1}^{n}F_{i}+F_{n+1}=F_{n+1}+F_{n+2}-1}$

and so we have the formula for ${\displaystyle n+1}$

$${\displaystyle \sum _{i=1}^{n+1}F_{i}=F_{n+3}-1}$$

Similarly, add ${\displaystyle F_{n+1}^{2}}$ to both sides of

$${\displaystyle \sum _{i=1}^{n}{F_{i}}^{2}=F_{n}F_{n+1}}$$

to give

$${\displaystyle \sum _{i=1}^{n}{F_{i}}^{2}+F_{n+1}^{2}=F_{n+1}\left(F_{n}+F_{n+1}\right)}$$

$${\displaystyle \sum _{i=1}^{n+1}{F_{i}}^{2}=F_{n+1}F_{n+2}}$$

Binet formula proofs [edit]

The Binet formula is

$${\displaystyle {\sqrt {5}}F_{n}=\varphi ^{n}-\psi ^{n}.}$$

This can be used to prove Fibonacci identities.

For example, to prove that ${\textstyle \sum _{i=1}^{n}F_{i}=F_{n+2}-1}$ note that the left hand side multiplied by ${\displaystyle {\sqrt {5}}}$ becomes

$${\displaystyle {\begin{aligned}1+&\varphi +\varphi ^{2}+\dots +\varphi ^{n}-\left(1+\psi +\psi ^{2}+\dots +\psi ^{n}\right)\\&={\frac {\varphi ^{n+1}-1}{\varphi -1}}-{\frac {\psi ^{n+1}-1}{\psi -1}}\\&={\frac {\varphi ^{n+1}-1}{-\psi }}-{\frac {\psi ^{n+1}-1}{-\varphi }}\\&={\frac {-\varphi ^{n+2}+\varphi +\psi ^{n+2}-\psi }{\varphi \psi }}\\&=\varphi ^{n+2}-\psi ^{n+2}-(\varphi -\psi )\\&={\sqrt {5}}(F_{n+2}-1)\\\end{aligned}}}$$

as required, using the facts ${\textstyle \varphi \psi =-1}$ and ${\textstyle \varphi -\psi ={\sqrt {5}}}$ to simplify the equations.

Other identities [edit]

Numerous other identities can be derived using various methods. Here are some of them:^[33]

Cassini's and Catalan's identities [edit]

Cassini's identity states that

$${\displaystyle {F_{n}}^{2}-F_{n+1}F_{n-1}=(-1)^{n-1}}$$

Catalan's identity is a generalization:

$${\displaystyle {F_{n}}^{2}-F_{n+r}F_{n-r}=(-1)^{n-r}F_{r}^{2}}$$

d'Ocagne's identity [edit]

$${\displaystyle F_{m}F_{n+1}-F_{m+1}F_{n}=(-1)^{n}F_{m-n}}$$

$${\displaystyle F_{2n}={F_{n+1}}^{2}-{F_{n-1}}^{2}=F_{n}\left(F_{n+1}+F_{n-1}\right)=F_{n}L_{n}}$$

where L _n is the n'th Lucas number. The last is an identity for doubling n; other identities of this type are

$${\displaystyle F_{3n}=2{F_{n}^{3}}+3F_{n}F_{n+1}F_{n-1}=5{F_{n}}^{3}+3(-1)^{n}F_{n}}$$

by Cassini's identity.

$${\displaystyle F_{3n+1}=F_{n+1}^{3}+3F_{n+1}{F_{n}}^{2}-F_{n}^{3}}$$

$${\displaystyle F_{3n+2}=F_{n+1}^{3}+3F_{n+1}^{2}F_{n}+{F_{n}}^{3}}$$

$${\displaystyle F_{4n}=4F_{n}F_{n+1}\left({F_{n+1}}^{2}+2{F_{n}}^{2}\right)-3{F_{n}}^{2}\left({F_{n}}^{2}+2{F_{n+1}}^{2}\right)}$$

These can be found experimentally using lattice reduction, and are useful in setting up the special number field sieve to factorize a Fibonacci number.

More generally,^[33]

$${\displaystyle F_{kn+c}=\sum _{i=0}^{k}{k \choose i}F_{c-i}F_{n}^{i}F_{n+1}^{k-i}.}$$

or alternatively

$${\displaystyle F_{kn+c}=\sum _{i=0}^{k}{k \choose i}F_{c+i}F_{n}^{i}F_{n-1}^{k-i}.}$$

Putting k = 2 in this formula, one gets again the formulas of the end of above section Matrix form.

Generating function [edit]

The generating function of the Fibonacci sequence is the power series

$${\displaystyle s(x)=\sum _{k=0}^{\infty }F_{k}x^{k}=\sum _{k=1}^{\infty }F_{k}x^{k}=0+x+x^{2}+2x^{3}+3x^{4}+\dots .}$$

This series is convergent for ${\displaystyle |x|<{\frac {1}{\varphi }},}$ and its sum has a simple closed-form:^[34]

$${\displaystyle s(x)={\frac {x}{1-x-x^{2}}}}$$

This can be proved by using the Fibonacci recurrence to expand each coefficient in the infinite sum:

$${\displaystyle {\begin{aligned}s(x)&=\sum _{k=0}^{\infty }F_{k}x^{k}\\&=F_{0}+F_{1}x+\sum _{k=2}^{\infty }F_{k}x^{k}\\&=F_{0}+F_{1}x+\sum _{k=2}^{\infty }\left(F_{k-1}+F_{k-2}\right)x^{k}\\&=x+\sum _{k=2}^{\infty }F_{k-1}x^{k}+\sum _{k=2}^{\infty }F_{k-2}x^{k}\\&=x+x\sum _{k=2}^{\infty }F_{k-1}x^{k-1}+x^{2}\sum _{k=2}^{\infty }F_{k-2}x^{k-2}\\&=x+x\sum _{k=1}^{\infty }F_{k}x^{k}+x^{2}\sum _{k=0}^{\infty }F_{k}x^{k}\\&=x+xs(x)+x^{2}s(x).\end{aligned}}}$$

Solving the equation

$${\displaystyle s(x)=x+xs(x)+x^{2}s(x)}$$

for s(x) results in the closed form.

${\displaystyle -s\left(-{\frac {1}{x}}\right)}$ gives the generating function for the negafibonacci numbers, and ${\displaystyle s(x)}$ satisfies the functional equation

$${\displaystyle s(x)=s\left(-{\frac {1}{x}}\right).}$$

The partial fraction decomposition is given by

$${\displaystyle s(x)={\frac {1}{\sqrt {5}}}\left({\frac {1}{1-\varphi x}}-{\frac {1}{1-\psi x}}\right)}$$

where ${\displaystyle \varphi ={\frac {1+{\sqrt {5}}}{2}}}$ is the golden ratio and ${\displaystyle \psi ={\frac {1-{\sqrt {5}}}{2}}}$ is its conjugate.

Reciprocal sums [edit]

Infinite sums over reciprocal Fibonacci numbers can sometimes be evaluated in terms of theta functions. For example, the sum of every odd-indexed reciprocal Fibonacci number can be written as

$${\displaystyle \sum _{k=1}^{\infty }{\frac {1}{F_{2k-1}}}={\frac {\sqrt {5}}{4}}\;\,\vartheta _{2}\!\left(0,{\frac {3-{\sqrt {5}}}{2}}\right)^{2},}$$

and the sum of squared reciprocal Fibonacci numbers as

$${\displaystyle \sum _{k=1}^{\infty }{\frac {1}{{F_{k}}^{2}}}={\frac {5}{24}}\left(\vartheta _{2}\!\left(0,{\frac {3-{\sqrt {5}}}{2}}\right)^{4}-\vartheta _{4}\!\left(0,{\frac {3-{\sqrt {5}}}{2}}\right)^{4}+1\right).}$$

If we add 1 to each Fibonacci number in the first sum, there is also the closed form

$${\displaystyle \sum _{k=1}^{\infty }{\frac {1}{1+F_{2k-1}}}={\frac {\sqrt {5}}{2}},}$$

and there is a nested sum of squared Fibonacci numbers giving the reciprocal of the golden ratio,

$${\displaystyle \sum _{k=1}^{\infty }{\frac {(-1)^{k+1}}{\sum _{j=1}^{k}{F_{j}}^{2}}}={\frac {{\sqrt {5}}-1}{2}}.}$$

The sum of all even-indexed reciprocal Fibonacci numbers is^[35]

$${\displaystyle \sum _{k=1}^{\infty }{\frac {1}{F_{2k}}}={\sqrt {5}}\left(L{\bigl (}\psi ^{2}{\bigr )}-L{\bigl (}\psi ^{4}{\bigr )}\right)}$$

with the Lambert series ${\displaystyle \textstyle L(q):=\sum _{k=1}^{\infty }{\frac {q^{k}}{1-q^{k}}},}$ since ${\displaystyle \textstyle {\frac {1}{F_{2k}}}={\sqrt {5}}\left({\frac {\psi ^{2k}}{1-\psi ^{2k}}}-{\frac {\psi ^{4k}}{1-\psi ^{4k}}}\right).}$

So the reciprocal Fibonacci constant is

$${\displaystyle \sum _{k=1}^{\infty }{\frac {1}{F_{k}}}=\sum _{k=1}^{\infty }{\frac {1}{F_{2k-1}}}+\sum _{k=1}^{\infty }{\frac {1}{F_{2k}}}=3.359885666243\dots }$$

Moreover, this number has been proved irrational by Richard André-Jeannin.^[36]

The Millin series gives the identity^[37]

$${\displaystyle \sum _{k=0}^{\infty }{\frac {1}{F_{2^{k}}}}={\frac {7-{\sqrt {5}}}{2}},}$$

which follows from the closed form for its partial sums as N tends to infinity:

$${\displaystyle \sum _{k=0}^{N}{\frac {1}{F_{2^{k}}}}=3-{\frac {F_{2^{N}-1}}{F_{2^{N}}}}.}$$

Primes and divisibility [edit]

Divisibility properties [edit]

Every third number of the sequence is even and more generally, every kth number of the sequence is a multiple of F_k . Thus the Fibonacci sequence is an example of a divisibility sequence. In fact, the Fibonacci sequence satisfies the stronger divisibility property^{[38] [39]}

$${\displaystyle \gcd(F_{m},F_{n})=F_{\gcd(m,n)}.}$$

Any three consecutive Fibonacci numbers are pairwise coprime, which means that, for every n,

gcd(F _n , F _n+1) = gcd(F _n , F _n+2) = gcd(F _n+1, F _n+2) = 1.

Every prime number p divides a Fibonacci number that can be determined by the value of p modulo 5. If p is congruent to 1 or 4 (mod 5), then p divides F _p − 1, and if p is congruent to 2 or 3 (mod 5), then, p divides F _p + 1. The remaining case is that p = 5, and in this case p divides F _p.

$${\displaystyle {\begin{cases}p=5&\Rightarrow p\mid F_{p},\\p\equiv \pm 1{\pmod {5}}&\Rightarrow p\mid F_{p-1},\\p\equiv \pm 2{\pmod {5}}&\Rightarrow p\mid F_{p+1}.\end{cases}}}$$

These cases can be combined into a single, non-piecewise formula, using the Legendre symbol:^[40]

$${\displaystyle p\mid F_{p-\left({\frac {5}{p}}\right)}.}$$

Primality testing [edit]

The above formula can be used as a primality test in the sense that if

$${\displaystyle n\mid F_{n-\left({\frac {5}{n}}\right)},}$$

where the Legendre symbol has been replaced by the Jacobi symbol, then this is evidence that n is a prime, and if it fails to hold, then n is definitely not a prime. If n is composite and satisfies the formula, then n is a Fibonacci pseudoprime. When m is large – say a 500-bit number – then we can calculate F _m (mod n) efficiently using the matrix form. Thus

$${\displaystyle {\begin{pmatrix}F_{m+1}&F_{m}\\F_{m}&F_{m-1}\end{pmatrix}}\equiv {\begin{pmatrix}1&1\\1&0\end{pmatrix}}^{m}{\pmod {n}}.}$$

Here the matrix power A ^m is calculated using modular exponentiation, which can be adapted to matrices.^[41]

Fibonacci primes [edit]

A Fibonacci prime is a Fibonacci number that is prime. The first few are:

2, 3, 5, 13, 89, 233, 1597, 28657, 514229, ... OEIS: A005478.

Fibonacci primes with thousands of digits have been found, but it is not known whether there are infinitely many.^[42]

F _kn is divisible by F _n , so, apart from F ₄ = 3, any Fibonacci prime must have a prime index. As there are arbitrarily long runs of composite numbers, there are therefore also arbitrarily long runs of composite Fibonacci numbers.

No Fibonacci number greater than F ₆ = 8 is one greater or one less than a prime number.^[43]

The only nontrivial square Fibonacci number is 144.^[44] Attila Pethő proved in 2001 that there is only a finite number of perfect power Fibonacci numbers.^[45] In 2006, Y. Bugeaud, M. Mignotte, and S. Siksek proved that 8 and 144 are the only such non-trivial perfect powers.^[46]

1, 3, 21, and 55 are the only triangular Fibonacci numbers, which was conjectured by Vern Hoggatt and proved by Luo Ming.^[47]

No Fibonacci number can be a perfect number.^[48] More generally, no Fibonacci number other than 1 can be multiply perfect,^[49] and no ratio of two Fibonacci numbers can be perfect.^[50]

Prime divisors [edit]

With the exceptions of 1, 8 and 144 (F ₁ = F ₂, F ₆ and F ₁₂) every Fibonacci number has a prime factor that is not a factor of any smaller Fibonacci number (Carmichael's theorem).^[51] As a result, 8 and 144 (F ₆ and F ₁₂) are the only Fibonacci numbers that are the product of other Fibonacci numbers OEIS: A235383.

The divisibility of Fibonacci numbers by a prime p is related to the Legendre symbol ${\displaystyle \left({\tfrac {p}{5}}\right)}$ which is evaluated as follows:

$${\displaystyle \left({\frac {p}{5}}\right)={\begin{cases}0&{\text{if }}p=5\\1&{\text{if }}p\equiv \pm 1{\pmod {5}}\\-1&{\text{if }}p\equiv \pm 2{\pmod {5}}.\end{cases}}}$$

If p is a prime number then

$${\displaystyle F_{p}\equiv \left({\frac {p}{5}}\right){\pmod {p}}\quad {\text{and}}\quad F_{p-\left({\frac {p}{5}}\right)}\equiv 0{\pmod {p}}.}$$

^{[52] [53]}

For example,

$${\displaystyle {\begin{aligned}({\tfrac {2}{5}})&=-1,&F_{3}&=2,&F_{2}&=1,\\({\tfrac {3}{5}})&=-1,&F_{4}&=3,&F_{3}&=2,\\({\tfrac {5}{5}})&=0,&F_{5}&=5,\\({\tfrac {7}{5}})&=-1,&F_{8}&=21,&F_{7}&=13,\\({\tfrac {11}{5}})&=+1,&F_{10}&=55,&F_{11}&=89.\end{aligned}}}$$

It is not known whether there exists a prime p such that

$${\displaystyle F_{p-\left({\frac {p}{5}}\right)}\equiv 0{\pmod {p^{2}}}.}$$

Such primes (if there are any) would be called Wall–Sun–Sun primes.

Also, if p ≠ 5 is an odd prime number then:^[54]

$${\displaystyle 5F_{\frac {p\pm 1}{2}}^{2}\equiv {\begin{cases}{\tfrac {1}{2}}\left(5\left({\frac {p}{5}}\right)\pm 5\right){\pmod {p}}&{\text{if }}p\equiv 1{\pmod {4}}\\{\tfrac {1}{2}}\left(5\left({\frac {p}{5}}\right)\mp 3\right){\pmod {p}}&{\text{if }}p\equiv 3{\pmod {4}}.\end{cases}}}$$

Example 1. p = 7, in this case p ≡ 3 (mod 4) and we have:

$${\displaystyle ({\tfrac {7}{5}})=-1:\qquad {\tfrac {1}{2}}\left(5({\tfrac {7}{5}})+3\right)=-1,\quad {\tfrac {1}{2}}\left(5({\tfrac {7}{5}})-3\right)=-4.}$$

$${\displaystyle F_{3}=2{\text{ and }}F_{4}=3.}$$

$${\displaystyle 5F_{3}^{2}=20\equiv -1{\pmod {7}}\;\;{\text{ and }}\;\;5F_{4}^{2}=45\equiv -4{\pmod {7}}}$$

Example 2. p = 11, in this case p ≡ 3 (mod 4) and we have:

$${\displaystyle ({\tfrac {11}{5}})=+1:\qquad {\tfrac {1}{2}}\left(5({\tfrac {11}{5}})+3\right)=4,\quad {\tfrac {1}{2}}\left(5({\tfrac {11}{5}})-3\right)=1.}$$

$${\displaystyle F_{5}=5{\text{ and }}F_{6}=8.}$$

$${\displaystyle 5F_{5}^{2}=125\equiv 4{\pmod {11}}\;\;{\text{ and }}\;\;5F_{6}^{2}=320\equiv 1{\pmod {11}}}$$

Example 3. p = 13, in this case p ≡ 1 (mod 4) and we have:

$${\displaystyle ({\tfrac {13}{5}})=-1:\qquad {\tfrac {1}{2}}\left(5({\tfrac {13}{5}})-5\right)=-5,\quad {\tfrac {1}{2}}\left(5({\tfrac {13}{5}})+5\right)=0.}$$

$${\displaystyle F_{6}=8{\text{ and }}F_{7}=13.}$$

$${\displaystyle 5F_{6}^{2}=320\equiv -5{\pmod {13}}\;\;{\text{ and }}\;\;5F_{7}^{2}=845\equiv 0{\pmod {13}}}$$

Example 4. p = 29, in this case p ≡ 1 (mod 4) and we have:

$${\displaystyle ({\tfrac {29}{5}})=+1:\qquad {\tfrac {1}{2}}\left(5({\tfrac {29}{5}})-5\right)=0,\quad {\tfrac {1}{2}}\left(5({\tfrac {29}{5}})+5\right)=5.}$$

$${\displaystyle F_{14}=377{\text{ and }}F_{15}=610.}$$

$${\displaystyle 5F_{14}^{2}=710645\equiv 0{\pmod {29}}\;\;{\text{ and }}\;\;5F_{15}^{2}=1860500\equiv 5{\pmod {29}}}$$

For odd n, all odd prime divisors of F _n are congruent to 1 modulo 4, implying that all odd divisors of F _n (as the products of odd prime divisors) are congruent to 1 modulo 4.^[55]

For example,

$${\displaystyle F_{1}=1,F_{3}=2,F_{5}=5,F_{7}=13,F_{9}=34=2\cdot 17,F_{11}=89,F_{13}=233,F_{15}=610=2\cdot 5\cdot 61.}$$

All known factors of Fibonacci numbers F(i) for all i < 50000 are collected at the relevant repositories.^{[56] [57]}

Periodicity modulo n [edit]

If the members of the Fibonacci sequence are taken modn, the resulting sequence is periodic with period at most6n.^[58] The lengths of the periods for various n form the so-called Pisano periods OEIS: A001175. Determining a general formula for the Pisano periods is an open problem, which includes as a subproblem a special instance of the problem of finding the multiplicative order of a modular integer or of an element in a finite field. However, for any particular n, the Pisano period may be found as an instance of cycle detection.

Magnitude [edit]

Since F_n is asymptotic to ${\displaystyle \varphi ^{n}/{\sqrt {5}}}$ , the number of digits in F _n is asymptotic to ${\displaystyle n\log _{10}\varphi \approx 0.2090\,n}$ . As a consequence, for every integer d > 1 there are either 4 or 5 Fibonacci numbers with d decimal digits.

More generally, in the base b representation, the number of digits in F _n is asymptotic to ${\displaystyle n\log _{b}\varphi .}$

Generalizations [edit]

The Fibonacci sequence is one of the simplest and earliest known sequences defined by a recurrence relation, and specifically by a linear difference equation. All these sequences may be viewed as generalizations of the Fibonacci sequence. In particular, Binet's formula may be generalized to any sequence that is a solution of a homogeneous linear difference equation with constant coefficients.

Some specific examples that are close, in some sense, from Fibonacci sequence include:

• Generalizing the index to negative integers to produce the negafibonacci numbers.
• Generalizing the index to real numbers using a modification of Binet's formula.^[33]
• Starting with other integers. Lucas numbers have L ₁ = 1, L ₂ = 3, and L_n = L _n−1 + L _n−2. Primefree sequences use the Fibonacci recursion with other starting points to generate sequences in which all numbers are composite.
• Letting a number be a linear function (other than the sum) of the 2 preceding numbers. The Pell numbers have P_n = 2P _{n − 1} + P _{n − 2}. If the coefficient of the preceding value is assigned a variable value x, the result is the sequence of Fibonacci polynomials.
• Not adding the immediately preceding numbers. The Padovan sequence and Perrin numbers have P(n) = P(n − 2) + P(n − 3).
• Generating the next number by adding 3 numbers (tribonacci numbers), 4 numbers (tetranacci numbers), or more. The resulting sequences are known as n-Step Fibonacci numbers.^[59]

Applications [edit]

Mathematics [edit]

The Fibonacci numbers are the sums of the "shallow" diagonals (shown in red) of Pascal's triangle.

The Fibonacci numbers occur in the sums of "shallow" diagonals in Pascal's triangle (see binomial coefficient):^[60]

The generating function can be expanded into

$${\displaystyle {\frac {x}{1-x-x^{2}}}=x+x^{2}(1+x)+x^{3}(1+x)^{2}+\dots +x^{k+1}(1+x)^{k}+\dots =\sum \limits _{n=0}^{\infty }F_{n}x^{n}}$$

and collecting like terms of ${\displaystyle x^{n}}$ , we have the identity

$${\displaystyle F_{n}=\sum _{k=0}^{\left\lfloor {\frac {n-1}{2}}\right\rfloor }{\binom {n-k-1}{k}}.}$$

To see how the formula is used, we can arrange the sums by the number of terms present:

5	= 1+1+1+1+1
	= 2+1+1+1	= 1+2+1+1	= 1+1+2+1	= 1+1+1+2
	= 2+2+1	= 2+1+2	= 1+2+2

which is ${\displaystyle {\binom {5}{0}}+{\binom {4}{1}}+{\binom {3}{2}}}$ , where we are choosing the positions of k twos from n-k-1 terms.

Use of the Fibonacci sequence to count {1, 2}-restricted compositions

These numbers also give the solution to certain enumerative problems,^[61] the most common of which is that of counting the number of ways of writing a given number n as an ordered sum of 1s and 2s (called compositions); there are F _n+1 ways to do this (equivalently, it's also the number of domino tilings of the ${\displaystyle 2\times n}$ rectangle). For example, there are F ₅₊₁ = F ₆ = 8 ways one can climb a staircase of 5 steps, taking one or two steps at a time:

5	= 1+1+1+1+1	= 2+1+1+1	= 1+2+1+1	= 1+1+2+1	= 2+2+1
	= 1+1+1+2	= 2+1+2	= 1+2+2

The figure shows that 8 can be decomposed into 5 (the number of ways to climb 4 steps, followed by a single-step) plus 3 (the number of ways to climb 3 steps, followed by a double-step). The same reasoning is applied recursively until a single step, of which there is only one way to climb.

The Fibonacci numbers can be found in different ways among the set of binary strings, or equivalently, among the subsets of a given set.

• The number of binary strings of length n without consecutive 1s is the Fibonacci number F _n+2 . For example, out of the 16 binary strings of length 4, there are F ₆ = 8 without consecutive 1s – they are 0000, 0001, 0010, 0100, 0101, 1000, 1001, and 1010. Equivalently, F _n+2 is the number of subsets S of {1, ..., n} without consecutive integers, that is, those S for which {i, i + 1} ⊈ S for every i . A bijection with the sums to n+1 is to replace 1 with 0 and 2 with 10, and drop the last zero.
• The number of binary strings of length n without an odd number of consecutive 1s is the Fibonacci number F _n+1 . For example, out of the 16 binary strings of length 4, there are F ₅ = 5 without an odd number of consecutive 1s – they are 0000, 0011, 0110, 1100, 1111. Equivalently, the number of subsets S of {1, ..., n} without an odd number of consecutive integers is F _n+1 . A bijection with the sums to n is to replace 1 with 0 and 2 with 11.
• The number of binary strings of length n without an even number of consecutive 0s or 1s is 2F _n . For example, out of the 16 binary strings of length 4, there are 2F ₄ = 6 without an even number of consecutive 0s or 1s – they are 0001, 0111, 0101, 1000, 1010, 1110. There is an equivalent statement about subsets.
• Yuri Matiyasevich was able to show that the Fibonacci numbers can be defined by a Diophantine equation, which led to his solving Hilbert's tenth problem.^[62]
• The Fibonacci numbers are also an example of a complete sequence. This means that every positive integer can be written as a sum of Fibonacci numbers, where any one number is used once at most.
• Moreover, every positive integer can be written in a unique way as the sum of one or more distinct Fibonacci numbers in such a way that the sum does not include any two consecutive Fibonacci numbers. This is known as Zeckendorf's theorem, and a sum of Fibonacci numbers that satisfies these conditions is called a Zeckendorf representation. The Zeckendorf representation of a number can be used to derive its Fibonacci coding.
• Starting with 5, every second Fibonacci number is the length of the hypotenuse of a right triangle with integer sides, or in other words, the largest number in a Pythagorean triple, obtained from the formula

  $${\displaystyle (F_{n}F_{n+3})^{2}+(2F_{n+1}F_{n+2})^{2}={F_{2n+3}}^{2}.}$$

  

  The sequence of Pythagorean triangles obtained from this formula has sides of lengths (3,4,5), (5,12,13), (16,30,34), (39,80,89), ... The middle side of each of these triangles is the sum of the three sides of the preceding triangle.^[63]
• The Fibonacci cube is an undirected graph with a Fibonacci number of nodes that has been proposed as a network topology for parallel computing.
• Fibonacci numbers appear in the ring lemma, used to prove connections between the circle packing theorem and conformal maps.^[64]

Computer science [edit]

Fibonacci tree of height 6. Balance factors green; heights red.
The keys in the left spine are Fibonacci numbers.

• The Fibonacci numbers are important in computational run-time analysis of Euclid's algorithm to determine the greatest common divisor of two integers: the worst case input for this algorithm is a pair of consecutive Fibonacci numbers.^[65]
• Fibonacci numbers are used in a polyphase version of the merge sort algorithm in which an unsorted list is divided into two lists whose lengths correspond to sequential Fibonacci numbers – by dividing the list so that the two parts have lengths in the approximate proportion φ. A tape-drive implementation of the polyphase merge sort was described in The Art of Computer Programming.
• A Fibonacci tree is a binary tree whose child trees (recursively) differ in height by exactly 1. So it is an AVL tree, and one with the fewest nodes for a given height — the "thinnest" AVL tree. These trees have a number of vertices that is a Fibonacci number minus one, an important fact in the analysis of AVL trees.^[66]
• Fibonacci numbers are used by some pseudorandom number generators.
• Fibonacci numbers arise in the analysis of the Fibonacci heap data structure.
• A one-dimensional optimization method, called the Fibonacci search technique, uses Fibonacci numbers.^[67]
• The Fibonacci number series is used for optional lossy compression in the IFF 8SVX audio file format used on Amiga computers. The number series compands the original audio wave similar to logarithmic methods such as μ-law.^{[68] [69]}
• They are also used in planning poker, which is a step in estimating in software development projects that use the Scrum methodology.

Nature [edit]

Yellow chamomile head showing the arrangement in 21 (blue) and 13 (aqua) spirals. Such arrangements involving consecutive Fibonacci numbers appear in a wide variety of plants.

Fibonacci sequences appear in biological settings,^[70] such as branching in trees, arrangement of leaves on a stem, the fruitlets of a pineapple,^[71] the flowering of artichoke, an uncurling fern and the arrangement of a pine cone,^[72] and the family tree of honeybees.^{[73] [74]} Kepler pointed out the presence of the Fibonacci sequence in nature, using it to explain the (golden ratio-related) pentagonal form of some flowers.^[75] Field daisies most often have petals in counts of Fibonacci numbers.^[76] In 1754, Charles Bonnet discovered that the spiral phyllotaxis of plants were frequently expressed in Fibonacci number series.^[77]

Przemysław Prusinkiewicz advanced the idea that real instances can in part be understood as the expression of certain algebraic constraints on free groups, specifically as certain Lindenmayer grammars.^[78]

Illustration of Vogel's model for n = 1 ... 500

A model for the pattern of florets in the head of a sunflower was proposed by Helmut Vogel in 1979.^[79] This has the form

$${\displaystyle \theta ={\frac {2\pi }{\varphi ^{2}}}n,\ r=c{\sqrt {n}}}$$

where n is the index number of the floret and c is a constant scaling factor; the florets thus lie on Fermat's spiral. The divergence angle, approximately 137.51°, is the golden angle, dividing the circle in the golden ratio. Because this ratio is irrational, no floret has a neighbor at exactly the same angle from the center, so the florets pack efficiently. Because the rational approximations to the golden ratio are of the form F(j):F(j + 1), the nearest neighbors of floret number n are those at n ± F(j) for some index j , which depends on r , the distance from the center. Sunflowers and similar flowers most commonly have spirals of florets in clockwise and counter-clockwise directions in the amount of adjacent Fibonacci numbers,^[80] typically counted by the outermost range of radii.^[81]

Fibonacci numbers also appear in the pedigrees of idealized honeybees, according to the following rules:

• If an egg is laid by an unmated female, it hatches a male or drone bee.
• If, however, an egg was fertilized by a male, it hatches a female.

Thus, a male bee always has one parent, and a female bee has two. If one traces the pedigree of any male bee (1 bee), he has 1 parent (1 bee), 2 grandparents, 3 great-grandparents, 5 great-great-grandparents, and so on. This sequence of numbers of parents is the Fibonacci sequence. The number of ancestors at each level, F _n , is the number of female ancestors, which is F _n−1 , plus the number of male ancestors, which is F _n−2 .^[82] This is under the unrealistic assumption that the ancestors at each level are otherwise unrelated.

The number of possible ancestors on the X chromosome inheritance line at a given ancestral generation follows the Fibonacci sequence. (After Hutchison, L. "Growing the Family Tree: The Power of DNA in Reconstructing Family Relationships".^[83])

It has been noticed that the number of possible ancestors on the human X chromosome inheritance line at a given ancestral generation also follows the Fibonacci sequence.^[83] A male individual has an X chromosome, which he received from his mother, and a Y chromosome, which he received from his father. The male counts as the "origin" of his own X chromosome ( ${\displaystyle F_{1}=1}$ ), and at his parents' generation, his X chromosome came from a single parent ( ${\displaystyle F_{2}=1}$ ). The male's mother received one X chromosome from her mother (the son's maternal grandmother), and one from her father (the son's maternal grandfather), so two grandparents contributed to the male descendant's X chromosome ( ${\displaystyle F_{3}=2}$ ). The maternal grandfather received his X chromosome from his mother, and the maternal grandmother received X chromosomes from both of her parents, so three great-grandparents contributed to the male descendant's X chromosome ( ${\displaystyle F_{4}=3}$ ). Five great-great-grandparents contributed to the male descendant's X chromosome ( ${\displaystyle F_{5}=5}$ ), etc. (This assumes that all ancestors of a given descendant are independent, but if any genealogy is traced far enough back in time, ancestors begin to appear on multiple lines of the genealogy, until eventually a population founder appears on all lines of the genealogy.)

The pathways of tubulins on intracellular microtubules arrange in patterns of 3, 5, 8 and 13.^[84]

Other [edit]

• In optics, when a beam of light shines at an angle through two stacked transparent plates of different materials of different refractive indexes, it may reflect off three surfaces: the top, middle, and bottom surfaces of the two plates. The number of different beam paths that have k reflections, for k > 1, is the ${\displaystyle k}$ th Fibonacci number. (However, when k = 1, there are three reflection paths, not two, one for each of the three surfaces.)^[85]
• Fibonacci retracement levels are widely used in technical analysis for financial market trading.
• Since the conversion factor 1.609344 for miles to kilometers is close to the golden ratio, the decomposition of distance in miles into a sum of Fibonacci numbers becomes nearly the kilometer sum when the Fibonacci numbers are replaced by their successors. This method amounts to a radix 2 number register in golden ratio base φ being shifted. To convert from kilometers to miles, shift the register down the Fibonacci sequence instead.^[86]
• Brasch et al. 2012 show how a generalised Fibonacci sequence also can be connected to the field of economics.^[87] In particular, it is shown how a generalised Fibonacci sequence enters the control function of finite-horizon dynamic optimisation problems with one state and one control variable. The procedure is illustrated in an example often referred to as the Brock–Mirman economic growth model.
• Mario Merz included the Fibonacci sequence in some of his artworks beginning in 1970.^[88]
• Joseph Schillinger (1895–1943) developed a system of composition which uses Fibonacci intervals in some of its melodies; he viewed these as the musical counterpart to the elaborate harmony evident within nature.^[89] See also Golden ratio § Music.

See also [edit]

• Elliott wave principle
• Embree–Trefethen constant
• The Fibonacci Association
• Fibonacci numbers in popular culture
• Fibonacci word
• Strong law of small numbers
• Verner Emil Hoggatt Jr.
• Wythoff array
• Fibonacci retracement

References [edit]

Footnotes

1. ^ "For four, variations of meters of two [and] three being mixed, five happens. For five, variations of two earlier – three [and] four, being mixed, eight is obtained. In this way, for six, [variations] of four [and] of five being mixed, thirteen happens. And like that, variations of two earlier meters being mixed, seven morae [is] twenty-one. In this way, the process should be followed in all mātrā-vṛttas" ^[14]

Citations

1. ^ Lucas 1891, p. 3.
2. ^ ^{a b} Sloane, N. J. A. (ed.). "Sequence A000045". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
3. ^ Beck & Geoghegan 2010.
4. ^ Bóna 2011, p. 180.
5. ^ Leonardo da Pisa (1202). File:Liber abbaci magliab f124r.jpg - Wikimedia Commons.
6. ^ ^{a b} Pisano 2002, pp. 404–05.
7. ^ ^{a b c} Goonatilake, Susantha (1998), Toward a Global Science, Indiana University Press, p. 126, ISBN978-0-253-33388-9
8. ^ ^{a b} Singh, Parmanand (1985), "The So-called Fibonacci numbers in ancient and medieval India", Historia Mathematica, 12 (3): 229–44, doi:10.1016/0315-0860(85)90021-7
9. ^ ^{a b} Knuth, Donald (2006), The Art of Computer Programming, 4. Generating All Trees – History of Combinatorial Generation, Addison–Wesley, p. 50, ISBN978-0-321-33570-8, it was natural to consider the set of all sequences of [L] and [S] that have exactly m beats. ...there are exactly Fm+1 of them. For example the 21 sequences when m = 7 are: [gives list]. In this way Indian prosodists were led to discover the Fibonacci sequence, as we have observed in Section 1.2.8 (from v.1)
10. ^ Knuth, Donald (1968), The Art of Computer Programming, 1, Addison Wesley, p. 100, ISBN978-81-7758-754-8, Before Fibonacci wrote his work, the sequence Fn had already been discussed by Indian scholars, who had long been interested in rhythmic patterns... both Gopala (before 1135 AD) and Hemachandra (c. 1150) mentioned the numbers 1,2,3,5,8,13,21 explicitly [see P. Singh Historia Math 12 (1985) 229–44]" p. 100 (3d ed)...
11. ^ ^{a b} Livio 2003, p. 197.
12. ^ Agrawala, VS (1969), Pāṇinikālīna Bhāratavarṣa (Hn.). Varanasi-I: TheChowkhamba Vidyabhawan, SadgurushiShya writes that Pingala was a younger brother of Pāṇini [Agrawala 1969, lb]. There is an alternative opinion that he was a maternal uncle of Pāṇini [Vinayasagar 1965, Preface, 121]. ... Agrawala [1969, 463–76], after a careful investigation, in which he considered the views of earlier scholars, has concluded that Pāṇini lived between 480 and 410 BC
13. ^ Singh, Parmanand (1985). "The So-called Fibonacci Numbers in Ancient and Medieval India" (PDF). Historia Mathematica. Academic Press. 12 (3): 232. doi:10.1016/0315-0860(85)90021-7.
14. ^ Velankar, HD (1962), 'Vṛttajātisamuccaya' of kavi Virahanka, Jodhpur: Rajasthan Oriental Research Institute, p. 101
15. ^ Livio 2003, p. 197–98.
16. ^ Shah, Jayant (1991). "A History of Piṅgala's Combinatorics" (PDF). Northeastern University: 41. Retrieved 4 January 2019.
17. ^ "Fibonacci's Liber Abaci (Book of Calculation)". The University of Utah. 13 December 2009. Retrieved 28 November 2018.
18. ^ Hemenway, Priya (2005). Divine Proportion: Phi In Art, Nature, and Science. New York: Sterling. pp. 20–21. ISBN1-4027-3522-7.
19. ^ Knott, Dr. Ron (25 September 2016). "The Fibonacci Numbers and Golden section in Nature – 1". University of Surrey . Retrieved 27 November 2018.
20. ^ Knott, Ron. "Fibonacci's Rabbits". University of Surrey Faculty of Engineering and Physical Sciences.
21. ^ Gardner, Martin (1996), Mathematical Circus, The Mathematical Association of America, p. 153, ISBN978-0-88385-506-5, It is ironic that Leonardo, who made valuable contributions to mathematics, is remembered today mainly because a 19th-century French number theorist, Édouard Lucas... attached the name Fibonacci to a number sequence that appears in a trivial problem in Liber abaci
22. ^ Knuth, Donald (2008-12-11), "Negafibonacci Numbers and the Hyperbolic Plane", Annual meeting, The Fairmont Hotel, San Jose, CA: The Mathematical Association of America
23. ^ Weisstein, Eric W. "Binet's Fibonacci Number Formula". MathWorld.
24. ^ Ball 2003, p. 156.
25. ^ Ball 2003, pp. 155–6.
26. ^ Kepler, Johannes (1966), A New Year Gift: On Hexagonal Snow, Oxford University Press, p. 92, ISBN978-0-19-858120-8
27. ^ Strena seu de Nive Sexangula, 1611
28. ^ Dijkstra, Edsger W. (1978), In honour of Fibonacci (PDF)
29. ^ Gessel, Ira (October 1972), "Fibonacci is a Square" (PDF), The Fibonacci Quarterly, 10 (4): 417–19, retrieved April 11, 2012
30. ^ Lucas 1891, p. 4.
31. ^ Vorobiev, Nikolaĭ Nikolaevich; Martin, Mircea (2002), "Chapter 1", Fibonacci Numbers, Birkhäuser, pp. 5–6, ISBN978-3-7643-6135-8
32. ^ Flajolet, Philippe; Sedgewick, Robert (2009). Analytic Combinatorics. Cambridge University Press. p. 42. ISBN978-0521898065.
33. ^ ^{a b c} Weisstein, Eric W. "Fibonacci Number". MathWorld.
34. ^ Glaister, P (1995), "Fibonacci power series", The Mathematical Gazette, 79 (486): 521–25, doi:10.2307/3618079, JSTOR 3618079
35. ^ Landau (1899) quoted according Borwein, Page 95, Exercise 3b.
36. ^ André-Jeannin, Richard (1989), "Irrationalité de la somme des inverses de certaines suites récurrentes", Comptes Rendus de l'Académie des Sciences, Série I, 308 (19): 539–41, MR 0999451
37. ^ Weisstein, Eric W. "Millin Series". MathWorld.
38. ^ Ribenboim, Paulo (2000), My Numbers, My Friends, Springer-Verlag
39. ^ Su, Francis E (2000), "Fibonacci GCD's, please", Mudd Math Fun Facts, et al, HMC, archived from the original on 2009-12-14, retrieved 2007-02-23
40. ^ Williams, H. C. (1982), "A note on the Fibonacci quotient ${\displaystyle F_{p-\varepsilon }/p}$ ", Canadian Mathematical Bulletin, 25 (3): 366–70, doi:10.4153/CMB-1982-053-0, hdl:10338.dmlcz/137492, MR 0668957 . Williams calls this property "well known".
41. ^ Prime Numbers, Richard Crandall, Carl Pomerance, Springer, second edition, 2005, p. 142.
42. ^ Weisstein, Eric W. "Fibonacci Prime". MathWorld.
43. ^ Honsberger, Ross (1985), "Mathematical Gems III", AMS Dolciani Mathematical Expositions (9): 133, ISBN978-0-88385-318-4
44. ^ Cohn, J. H. E. (1964). "On square Fibonacci numbers". The Journal of the London Mathematical Society. 39: 537–540. doi:10.1112/jlms/s1-39.1.537. MR 0163867.
45. ^ Pethő, Attila (2001), "Diophantine properties of linear recursive sequences II", Acta Mathematica Academiae Paedagogicae Nyíregyháziensis, 17: 81–96
46. ^ Bugeaud, Y; Mignotte, M; Siksek, S (2006), "Classical and modular approaches to exponential Diophantine equations. I. Fibonacci and Lucas perfect powers", Ann. Math., 2 (163): 969–1018, arXiv:math/0403046, Bibcode:2004math......3046B, doi:10.4007/annals.2006.163.969, S2CID 10266596
47. ^ Ming, Luo (1989), "On triangular Fibonacci numbers" (PDF), Fibonacci Quart., 27 (2): 98–108
48. ^ Luca, Florian (2000). "Perfect Fibonacci and Lucas numbers". Rendiconti del Circolo Matematico di Palermo. 49 (2): 313–18. doi:10.1007/BF02904236. ISSN 1973-4409. MR 1765401. S2CID 121789033.
49. ^ Broughan, Kevin A.; González, Marcos J.; Lewis, Ryan H.; Luca, Florian; Mejía Huguet, V. Janitzio; Togbé, Alain (2011). "There are no multiply-perfect Fibonacci numbers". Integers. 11a: A7. MR 2988067.
50. ^ Luca, Florian; Mejía Huguet, V. Janitzio (2010). "On Perfect numbers which are ratios of two Fibonacci numbers". Annales Mathematicae at Informaticae. 37: 107–24. ISSN 1787-6117. MR 2753031.
51. ^ Knott, Ron, The Fibonacci numbers, UK: Surrey
52. ^ Ribenboim, Paulo (1996), The New Book of Prime Number Records, New York: Springer, p. 64, ISBN978-0-387-94457-9
53. ^ Lemmermeyer 2000, pp. 73–74, ex. 2.25–28.
54. ^ Lemmermeyer 2000, pp. 73–74, ex. 2.28.
55. ^ Lemmermeyer 2000, p. 73, ex. 2.27.
56. ^ Fibonacci and Lucas factorizations, Mersennus collects all known factors of F(i) with i < 10000.
57. ^ Factors of Fibonacci and Lucas numbers, Red golpe collects all known factors of F(i) with 10000 < i < 50000.
58. ^ Freyd, Peter; Brown, Kevin S. (1993), "Problems and Solutions: Solutions: E3410", The American Mathematical Monthly, 99 (3): 278–79, doi:10.2307/2325076, JSTOR 2325076
59. ^ Weisstein, Eric W. "Fibonacci n-Step Number". MathWorld.
60. ^ Lucas 1891, p. 7.
61. ^ Stanley, Richard (2011). Enumerative Combinatorics I (2nd ed.). Cambridge Univ. Press. p. 121, Ex 1.35. ISBN978-1-107-60262-5.
62. ^ Harizanov, Valentina (1995), "Review of Yuri V. Matiyasevich, Hibert's Tenth Problem", Modern Logic, 5 (3): 345–55 .
63. ^ Pagni, David (September 2001), "Fibonacci Meets Pythagoras", Mathematics in School, 30 (4): 39–40, JSTOR 30215477
64. ^ Stephenson, Kenneth (2005), Introduction to Circle Packing: The Theory of Discrete Analytic Functions, Cambridge University Press, ISBN978-0-521-82356-2, MR 2131318 ; see especially Lemma 8.2 (Ring Lemma), pp. 73–74, and Appendix B, The Ring Lemma, pp. 318–321.
65. ^ Knuth, Donald E (1997), The Art of Computer Programming, 1: Fundamental Algorithms (3rd ed.), Addison–Wesley, p. 343, ISBN978-0-201-89683-1
66. ^ Adelson-Velsky, Georgy; Landis, Evgenii (1962). "An algorithm for the organization of information". Proceedings of the USSR Academy of Sciences (in Russian). 146: 263–266. English translation by Myron J. Ricci in Soviet Mathematics - Doklady, 3:1259–1263, 1962.
67. ^ Avriel, M; Wilde, DJ (1966), "Optimality of the Symmetric Fibonacci Search Technique", Fibonacci Quarterly (3): 265–69
68. ^ Amiga ROM Kernel Reference Manual, Addison–Wesley, 1991
69. ^ "IFF", Multimedia Wiki
70. ^ Douady, S; Couder, Y (1996), "Phyllotaxis as a Dynamical Self Organizing Process" (PDF), Journal of Theoretical Biology, 178 (3): 255–74, doi:10.1006/jtbi.1996.0026, archived from the original (PDF) on 2006-05-26
71. ^ Jones, Judy; Wilson, William (2006), "Science", An Incomplete Education, Ballantine Books, p. 544, ISBN978-0-7394-7582-9
72. ^ Brousseau, A (1969), "Fibonacci Statistics in Conifers", Fibonacci Quarterly (7): 525–32
73. ^ "Marks for the da Vinci Code: B–". Maths. Computer Science For Fun: CS4FN.
74. ^ Scott, T.C.; Marketos, P. (March 2014), On the Origin of the Fibonacci Sequence (PDF), MacTutor History of Mathematics archive, University of St Andrews
75. ^ Livio 2003, p. 110.
76. ^ Livio 2003, pp. 112–13.
77. ^ "The Secret of the Fibonacci Sequence in Trees". American Museum of Natural History. 2011. Archived from the original on 4 May 2013. Retrieved 4 February 2019.
78. ^ Prusinkiewicz, Przemyslaw; Hanan, James (1989), Lindenmayer Systems, Fractals, and Plants (Lecture Notes in Biomathematics), Springer-Verlag, ISBN978-0-387-97092-9
79. ^ Vogel, Helmut (1979), "A better way to construct the sunflower head", Mathematical Biosciences, 44 (3–4): 179–89, doi:10.1016/0025-5564(79)90080-4
80. ^ Livio 2003, p. 112.
81. ^ Prusinkiewicz, Przemyslaw; Lindenmayer, Aristid (1990), "4", The Algorithmic Beauty of Plants, Springer-Verlag, pp. 101–107, ISBN978-0-387-97297-8
82. ^ "The Fibonacci sequence as it appears in nature" (PDF), The Fibonacci Quarterly, 1 (1): 53–56, 1963
83. ^ ^{a b} Hutchison, Luke (September 2004). "Growing the Family Tree: The Power of DNA in Reconstructing Family Relationships" (PDF). Proceedings of the First Symposium on Bioinformatics and Biotechnology (BIOT-04) . Retrieved 2016-09-03 .
84. ^ Hameroff, Stuart; Penrose, Roger (March 2014). "Consciousness in the universe: A review of the 'Orch OR' theory". Physics of Life Reviews. Elsevier. 11 (1): 39–78. Bibcode:2014PhLRv..11...39H. doi:10.1016/j.plrev.2013.08.002. PMID 24070914.
85. ^ Livio 2003, pp. 98–99.
86. ^ "Zeckendorf representation", Encyclopedia of Math
87. ^ Brasch, T. von; Byström, J.; Lystad, L.P. (2012), "Optimal Control and the Fibonacci Sequence", Journal of Optimization Theory and Applications, 154 (3): 857–78, doi:10.1007/s10957-012-0061-2, hdl:11250/180781, S2CID 8550726
88. ^ Livio 2003, p. 176.
89. ^ Livio 2003, p. 193.

Works cited [edit]

• Ball, Keith M (2003), "8: Fibonacci's Rabbits Revisited", Strange Curves, Counting Rabbits, and Other Mathematical Explorations, Princeton, NJ: Princeton University Press, ISBN978-0-691-11321-0 .
• Beck, Matthias; Geoghegan, Ross (2010), The Art of Proof: Basic Training for Deeper Mathematics, New York: Springer, ISBN978-1-4419-7022-0 .
• Bóna, Miklós (2011), A Walk Through Combinatorics (3rd ed.), New Jersey: World Scientific, ISBN978-981-4335-23-2 .
• Bóna, Miklós (2016), A Walk Through Combinatorics (4th Revised ed.), New Jersey: World Scientific, ISBN978-981-3148-84-0 .
• Borwein, Jonathan M.; Borwein, Peter B. (July 1998), Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity, Wiley, pp. 91–101, ISBN978-0-471-31515-5
• Lemmermeyer, Franz (2000), Reciprocity Laws: From Euler to Eisenstein, Springer Monographs in Mathematics, New York: Springer, ISBN978-3-540-66957-9 .
• Livio, Mario (2003) [2002]. The Golden Ratio: The Story of Phi, the World's Most Astonishing Number (First trade paperback ed.). New York City: Broadway Books. ISBN0-7679-0816-3.
• Lucas, Édouard (1891), Théorie des nombres (in French), 1, Paris: Gauthier-Villars, https://books.google.com/books?id=_hsPAAAAIAAJ .
• Pisano, Leonardo (2002), Fibonacci's Liber Abaci: A Translation into Modern English of the Book of Calculation, Sources and Studies in the History of Mathematics and Physical Sciences, Sigler, Laurence E, trans, Springer, ISBN978-0-387-95419-6

External links [edit]

• Periods of Fibonacci Sequences Mod m at MathPages
• Scientists find clues to the formation of Fibonacci spirals in nature
• Fibonacci Sequence on In Our Time at the BBC
• "Fibonacci numbers", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• OEIS sequence A000045 (Fibonacci numbers)

Ross Honsberger Mathematical Gems Pdf

Source: https://en.wikipedia.org/wiki/Fibonacci_number

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