In mathematics, the Fibonacci sequence is a
sequence in which each number is the sum of the two preceding ones. Numbers that are part of the Fibonacci sequence are known as Fibonacci numbers, commonly denoted F_{n}. The sequence commonly starts from 0 and 1, although some authors start the sequence from 1 and 1 or sometimes (as did Fibonacci) from 1 and 2. Starting from 0 and 1, the sequence begins
The Fibonacci numbers were first described in
Indian mathematics as early as 200 BC in work by
Pingala on enumerating possible patterns of
Sanskrit poetry formed from syllables of two lengths.^{
[2]}^{
[3]}^{
[4]} They are named after the Italian mathematician Leonardo of Pisa, also known as
Fibonacci, who introduced the sequence to Western European mathematics in his 1202 book Liber Abaci.^{
[5]}
Fibonacci numbers are also 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 also closely related to
Lucas numbers, which obey the same
recurrence relation and with the Fibonacci numbers form a complementary pair of
Lucas sequences.
Under some older definitions, the value $F_{0}=0$ is omitted, so that the sequence starts with $F_{1}=F_{2}=1,$ and the recurrence $F_{n}=F_{n-1}+F_{n-2}$ is valid for n > 2.^{
[7]}^{
[8]}
The Fibonacci sequence appears in
Indian mathematics, in connection with
Sanskrit prosody.^{
[3]}^{
[9]}^{
[10]} 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}.^{
[4]}
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.^{
[11]}Bharata Muni also expresses knowledge of the sequence in the Natya Shastra (c. 100 BC–c. 350 AD).^{
[12]}^{
[2]}
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):^{
[10]}
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,^{
[2]} writing that "the sum of the last and the one before the last is the number ... of the next mātrā-vṛtta."^{
[14]}^{
[15]}
Europe
The Fibonacci sequence first appears in the book Liber Abaci (The Book of Calculation, 1202) by
Fibonacci^{
[16]}^{
[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 to gestate for a month, 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 nth 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 nth month is the nth Fibonacci number.^{
[20]}
The name "Fibonacci sequence" was first used by the 19th-century number theorist
Édouard Lucas.^{
[21]}
To see the relation between the sequence and these constants,^{
[24]} note that φ and ψ are both solutions of the equation ${\textstyle x^{2}=x+1}$ and thus $x^{n}=x^{n-1}+x^{n-2},$ so the powers of φ and ψ satisfy the Fibonacci recursion. In other words,
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:
Since
${\textstyle \left|{\frac {\psi ^{n}}{\sqrt {5}}}\right|<{\frac {1}{2}}}$ for all n ≥ 0, the number F_{n} is the closest
integer to ${\frac {\varphi ^{n}}{\sqrt {5}}}$. Therefore, it can be found by
rounding, using the nearest integer function:
In fact, the rounding error is very small, being less than 0.1 for n ≥ 4, and less than 0.01 for n ≥ 8. This formula is easily inverted to find an index of a Fibonacci number F:
where $\log _{\varphi }(x)=\ln(x)/\ln(\varphi )=\log _{10}(x)/\log _{10}(\varphi )$, $\ln(\varphi )=0.481211\ldots$,^{
[25]} and $\log _{10}(\varphi )=0.208987\ldots$.^{
[26]}
Magnitude
Since F_{n} is
asymptotic to $\varphi ^{n}/{\sqrt {5}}$, the number of digits in F_{n} is asymptotic to $n\log _{10}\varphi \approx 0.2090\,n$. As a consequence, for every integer d > 1 there are either 4 or 5 Fibonacci numbers with d decimal digits.
More generally, in the
baseb representation, the number of digits in F_{n} is asymptotic to $n\log _{b}\varphi ={\frac {n\log \varphi }{\log b}}.$
Limit of consecutive quotients
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 $\varphi \colon$^{
[27]}^{
[28]}
This convergence holds regardless of the starting values $U_{0}$ and $U_{1}$, unless $U_{1}=-U_{0}/\varphi$. 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.
In general, $\lim _{n\to \infty }{\frac {F_{n+m}}{F_{n}}}=\varphi ^{m}$, because the ratios between consecutive Fibonacci numbers approaches $\varphi$.
Decomposition of powers
Since the golden ratio satisfies the equation
$\varphi ^{2}=\varphi +1,$
this expression can be used to decompose higher powers $\varphi ^{n}$ as a linear function of lower powers, which in turn can be decomposed all the way down to a linear combination of $\varphi$ and 1. The resulting
recurrence relationships yield Fibonacci numbers as the linear
coefficients:
For $\psi =-1/\varphi$, it is also the case that $\psi ^{2}=\psi +1$ and it is also the case that
$\psi ^{n}=F_{n}\psi +F_{n-1}.$
These expressions are also true for n < 1 if the Fibonacci sequence F_{n} is
extended to negative integers using the Fibonacci rule $F_{n}=F_{n+2}-F_{n+1}.$
Identification
Binet's formula provides a proof that a positive integer x is a Fibonacci number
if and only if at least one of $5x^{2}+4$ or $5x^{2}-4$ is a
perfect square.^{
[29]} This is because Binet's formula, which can be written as $F_{n}=(\varphi ^{n}-(-1)^{n}\varphi ^{-n})/{\sqrt {5}}$, can be multiplied by ${\sqrt {5}}\varphi ^{n}$ and solved as a
quadratic equation in $\varphi ^{n}$ via the
quadratic formula:
which yields ${\vec {F}}_{n}=\mathbf {A} ^{n}{\vec {F}}_{0}$. The
eigenvalues of the
matrixA are $\varphi ={\frac {1}{2}}(1+{\sqrt {5}})$ and $\psi =-\varphi ^{-1}={\frac {1}{2}}(1-{\sqrt {5}})$ corresponding to the respective
eigenvectors
where $\Lambda ={\begin{pmatrix}\varphi &0\\0&-\varphi ^{-1}\end{pmatrix}}$ and $S={\begin{pmatrix}\varphi &-\varphi ^{-1}\\1&1\end{pmatrix}}.$
The closed-form expression for the nth element in the Fibonacci series is therefore given by
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:
For a given n, this matrix can be computed in O(log(n)) arithmetic operations, using the
exponentiation by squaring method.
Taking the determinant of both sides of this equation yields
Cassini's identity,
$(-1)^{n}=F_{n+1}F_{n-1}-{F_{n}}^{2}.$
Moreover, since A^{n}A^{m} = A^{n+m} for any
square matrixA, 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),
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 nth 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).^{
[30]}
Combinatorial identities
Combinatorial proofs
Most identities involving Fibonacci numbers can be proved using
combinatorial arguments using the fact that $F_{n}$ can be interpreted as the number of (possibly empty) sequences of 1s and 2s whose sum is $n-1$. This can be taken as the definition of $F_{n}$ with the conventions $F_{0}=0$, meaning no such sequence exists whose sum is −1, and $F_{1}=1$, meaning the empty sequence "adds up" to 0. In the following, $|{...}|$ is the
cardinality of a
set:
Excluding the first element, the remaining terms in each sequence sum to $n-2$ or $n-3$ and the cardinality of each set is $F_{n-1}$ or $F_{n-2}$ giving a total of $F_{n-1}+F_{n-2}$ sequences, showing this is equal to $F_{n}$.
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.^{
[31]} In symbols:
$\sum _{i=1}^{n}F_{i}=F_{n+2}-1$
This may be seen by dividing all sequences summing to $n+1$ based on the location of the first 2. Specifically, each set consists of those sequences that start $\{2,...\},\{1,2,...\},...,$ until the last two sets $\{\{1,1,...,1,2\}\},\{\{1,1,...,1\}\}$ each with cardinality 1.
Following the same logic as before, by summing the cardinality of each set we see that
... where the last two terms have the value $F_{1}=1$. From this it follows that $\sum _{i=1}^{n}F_{i}=F_{n+2}-1$.
A similar argument, grouping the sums by the position of the first 1 rather than the first 2 gives two more identities:
$\sum _{i=0}^{n-1}F_{2i+1}=F_{2n}$
and
$\sum _{i=1}^{n}F_{2i}=F_{2n+1}-1.$
In words, the sum of the first Fibonacci numbers with
odd index up to $F_{2n-1}$ is the (2n)th Fibonacci number, and the sum of the first Fibonacci numbers with
even index up to $F_{2n}$ is the (2n + 1)st Fibonacci number minus 1.^{
[32]}
A different trick may be used to prove
$\sum _{i=1}^{n}F_{i}^{2}=F_{n}F_{n+1}$
or in words, the sum of the squares of the first Fibonacci numbers up to $F_{n}$ is the product of the nth and (n + 1)st Fibonacci numbers. To see this, begin with a Fibonacci rectangle of size $F_{n}\times F_{n+1}$ and decompose it into squares of size $F_{n},F_{n-1},...,F_{1}$; from this the identity follows by comparing
areas:
Symbolic method
The sequence $(F_{n})_{n\in \mathbb {N} }$ is also considered using the
symbolic method.^{
[33]} More precisely, this sequence corresponds to a
specifiable combinatorial class. The specification of this sequence is $\operatorname {Seq} ({\mathcal {Z+Z^{2}}})$. Indeed, as stated above, the $n$-th Fibonacci number equals the number of
combinatorial compositions (ordered
partitions) of $n-1$ using terms 1 and 2.
where ${\textstyle \varphi ={\tfrac {1}{2}}\left(1+{\sqrt {5}}\right)}$ is the golden ratio and $\psi ={\tfrac {1}{2}}\left(1-{\sqrt {5}}\right)$ is its
conjugate.
The related function ${\textstyle z\mapsto -s\left(-1/z\right)}$ is the generating function for the
negafibonacci numbers, and $s(z)$ satisfies the
functional equation
$s(z)=s\!\left(-{\frac {1}{z}}\right).$
Using $z$ equal to any of 0.01, 0.001, 0.0001, etc. lays out the first Fibonacci numbers in the decimal expansion of $s(z)$. For example, $s(0.001)={\frac {0.001}{0.998999}}=0.001001002003005008013021....$
Reciprocal sums
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
Every third number of the sequence is even (a multiple of $F_{3}=2$) 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^{
[40]}^{
[41]}
Every
prime numberp divides a Fibonacci number that can be determined by the value of pmodulo 5. If p is congruent to 1 or 4 modulo 5, then p divides F_{p−1}, and if p is congruent to 2 or 3 modulo 5, then, p divides F_{p+1}. The remaining case is that p = 5, and in this case p divides F_{p}.
These cases can be combined into a single, non-
piecewise formula, using the
Legendre symbol:^{
[42]}
$p\mid F_{p\;-\,\left({\frac {5}{p}}\right)}.$
Primality testing
The above formula can be used as a
primality test in the sense that if
$n\mid F_{n\;-\,\left({\frac {5}{n}}\right)},$
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
A Fibonacci prime is a Fibonacci number that is
prime. The first few are:^{
[44]}
2, 3, 5, 13, 89, 233, 1597, 28657, 514229, ...
Fibonacci primes with thousands of digits have been found, but it is not known whether there are infinitely many.^{
[45]}
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.^{
[46]}
The only nontrivial
square Fibonacci number is 144.^{
[47]} Attila Pethő proved in 2001 that there is only a finite number of
perfect power Fibonacci numbers.^{
[48]} In 2006, Y. Bugeaud, M. Mignotte, and S. Siksek proved that 8 and 144 are the only such non-trivial perfect powers.^{
[49]}
No Fibonacci number can be a
perfect number.^{
[51]} More generally, no Fibonacci number other than 1 can be
multiply perfect,^{
[52]} and no ratio of two Fibonacci numbers can be perfect.^{
[53]}
Prime divisors
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).^{
[54]} As a result, 8 and 144 (F_{6} and F_{12}) are the only Fibonacci numbers that are the product of other Fibonacci numbers.^{
[55]}
The divisibility of Fibonacci numbers by a prime p is related to the
Legendre symbol$\left({\tfrac {p}{5}}\right)$ which is evaluated as follows:
$5{F_{14}}^{2}=710645\equiv 0{\pmod {29}}\;\;{\text{ and }}\;\;5{F_{15}}^{2}=1860500\equiv 5{\pmod {29}}$
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.^{
[59]}
If the members of the Fibonacci sequence are taken mod n, the resulting sequence is
periodic with period at most 6n.^{
[62]} The lengths of the periods for various n form the so-called
Pisano periods.^{
[63]} 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.
Some specific examples that are close, in some sense, to the 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.^{
[34]}
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.
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.^{
[64]}
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 ${\binom {5}{0}}+{\binom {4}{1}}+{\binom {3}{2}}$, where we are choosing the positions of k twos from n−k−1 terms.
These numbers also give the solution to certain enumerative problems,^{
[66]} 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 $2\times n$ 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
binarystrings, 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. Such strings are the binary representations of
Fibbinary numbers. 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.
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
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.^{
[68]}
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.^{
[71]}
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.^{
[72]}
The Fibonacci number series is used for optional
lossy compression in the
IFF8SVX audio file format used on
Amiga computers. The number series
compands the original audio wave similar to logarithmic methods such as
μ-law.^{
[73]}^{
[74]}
Some Agile teams use a modified series called the "Modified Fibonacci Series" in
planning poker, as an estimation tool. Planning Poker is a formal part of the
Scaled Agile Framework.^{
[75]}
Fibonacci sequences appear in biological settings,^{
[76]} such as branching in trees,
arrangement of leaves on a stem, the fruitlets of a
pineapple,^{
[77]} the flowering of
artichoke, the arrangement of a
pine cone,^{
[78]} and the family tree of
honeybees.^{
[79]}^{
[80]}Kepler pointed out the presence of the Fibonacci sequence in nature, using it to explain the (
golden ratio-related)
pentagonal form of some flowers.^{
[81]} Field
daisies most often have petals in counts of Fibonacci numbers.^{
[82]} In 1830, K. F. Schimper and A. Braun discovered that the
parastichies (spiral
phyllotaxis) of plants were frequently expressed as fractions involving Fibonacci numbers.^{
[83]}
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,^{
[86]} typically counted by the outermost range of radii.^{
[87]}
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}.^{
[88]} This is under the unrealistic assumption that the ancestors at each level are otherwise unrelated.
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.^{
[89]} 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 ($F_{1}=1$), and at his parents' generation, his X chromosome came from a single parent ($F_{2}=1$). 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 ($F_{3}=2$). 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 ($F_{4}=3$). Five great-great-grandparents contributed to the male descendant's X chromosome ($F_{5}=5$), 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.)
Other
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 $k$th Fibonacci number. (However, when k = 1, there are three reflection paths, not two, one for each of the three surfaces.)^{
[90]}
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 baseφ being shifted. To convert from kilometers to miles, shift the register down the Fibonacci sequence instead.^{
[91]}
The measured values of voltages and currents in the infinite resistor chain circuit (also called the
resistor ladder or infinite series-parallel circuit) follow the Fibonacci sequence. The intermediate results of adding the alternating series and parallel resistances yields fractions composed of consecutive Fibonacci numbers. The equivalent resistance of the entire circuit equals the golden ratio.^{
[92]}
Brasch et al. 2012 show how a generalized Fibonacci sequence also can be connected to the field of
economics.^{
[93]} In particular, it is shown how a generalized 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.^{
[94]}
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.^{
[95]} See also
Golden ratio § Music.
Wythoff array – Infinite matrix of integers derived from the Fibonacci sequence
References
Explanatory footnotes
^"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" ^{
[13]}
^
^{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
^
^{a}^{b}Knuth, Donald (2006),
The Art of Computer Programming, vol. 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)
^Knuth, Donald (1968),
The Art of Computer Programming, vol. 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)...
^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
^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
^Beutelspacher, Albrecht; Petri, Bernhard (1996), "Fibonacci-Zahlen", Der Goldene Schnitt, Vieweg+Teubner Verlag, pp. 87–98,
doi:
10.1007/978-3-322-85165-9_6
^Honsberger, Ross (1985),
"Millin's series", Mathematical Gems III, Dolciani Mathematical Expositions, vol. 9, American Mathematical Society, pp. 135–136,
ISBN9781470457181
^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,
MR0163867
^Pethő, Attila (2001), "Diophantine properties of linear recursive sequences II", Acta Mathematica Academiae Paedagogicae Nyíregyháziensis, 17: 81–96
^Freyd, Peter; Brown, Kevin S. (1993), "Problems and Solutions: Solutions: E3410", The American Mathematical Monthly, 99 (3): 278–79,
doi:
10.2307/2325076,
JSTOR2325076
^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.
^Avriel, M; Wilde, DJ (1966), "Optimality of the Symmetric Fibonacci Search Technique", Fibonacci Quarterly (3): 265–69
^Amiga ROM Kernel Reference Manual, Addison–Wesley, 1991
^Varenne, Franck (2010),
Formaliser le vivant - Lois, Théories, Modèles (in French), Hermann, p. 28,
ISBN9782705678128, retrieved 2022-10-30, En 1830, K. F. Schimper et A. Braun [...]. Ils montraient que si l'on représente cet angle de divergence par une fraction reflétant le nombre de tours par feuille ([...]), on tombe régulièrement sur un des nombres de la suite de Fibonacci pour le numérateur [...].
^Prusinkiewicz, Przemyslaw; Hanan, James (1989), Lindenmayer Systems, Fractals, and Plants (Lecture Notes in Biomathematics),
Springer-Verlag,
ISBN978-0-387-97092-9
^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
^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,
S2CID8550726
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.
Lucas, Édouard (1891),
Théorie des nombres (in French), vol. 1, Paris: Gauthier-Villars.
Sigler, L. E. (2002), Fibonacci's Liber Abaci: A Translation into Modern English of Leonardo Pisano's Book of Calculation, Sources and Studies in the History of Mathematics and Physical Sciences, Springer,
ISBN978-0-387-95419-6