In mathematics, the Fibonacci sequence is a sequence in which each element is the sum of the two elements that precede it. Numbers that are part of the Fibonacci sequence are known as Fibonacci numbers, commonly denoted Fn . Many writers begin the sequence with 0 and 1, although some authors start it from 1 and 1[1][2] and some (as did Fibonacci) from 1 and 2. Starting from 0 and 1, the sequence begins
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ... (sequence A000045 in the OEIS) A tiling with squares whose side lengths are successive Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13 and 21The 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.[3][4][5] 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.[6]
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 lees on a stem, the fruit sprouts of a pineapple, the flowering of an artichoke, and the arrangement of a pine cone's bracts, though they do not occur in all species.
Fibonacci numbers are also strongly related to the golden ratio: Binet's formula expresses the n-th 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.
Contents 1 Definition 2 History 2.1 India 2.2 Europe 3 Relation to the golden ratio 3.1 Closed-form expression 3.2 Computation by rounding 3.3 Magnitude 3.4 Limit of consecutive quotients 3.5 Decomposition of powers 3.6 Identification 4 Matrix form 5 Combinatorial identities 5.1 Combinatorial proofs 5.2 Induction proofs 5.3 Binet formula proofs 6 Other identities 6.1 Cassini's and Catalan's identities 6.2 d'Ocagne's identity 7 Generating function 8 Reciprocal sums 9 Primes and divisibility 9.1 Divisibility properties 9.2 Primality testing 9.3 Fibonacci primes 9.4 Prime divisors 9.5 Periodicity modulo n 10 Generalizations 11 Applications 11.1 Mathematics 11.2 Computer science 11.3 Nature 11.4 Other 12 See also 13 References 13.1 Explanatory footnotes 13.2 Citations 13.3 Works cited 14 External links Definition edit 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)The Fibonacci numbers may be defined by the recurrence relation[7] F 0 = 0 , F 1 = 1 , {\displaystyle F_{0}=0,\quad F_{1}=1,} and F n = F n − 1 + F n − 2 {\displaystyle F_{n}=F_{n-1}+F_{n-2}} for n > 1.
Under some older definitions, the value F 0 = 0 {\displaystyle F_{0}=0} is omitted, so that the sequence starts with F 1 = F 2 = 1 , {\displaystyle F_{1}=F_{2}=1,} and the recurrence F n = F n − 1 + F n − 2 {\displaystyle F_{n}=F_{n-1}+F_{n-2}} is valid for n > 2.[8][9]
The first 20 Fibonacci numbers Fn are:
F0 F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F13 F14 F15 F16 F17 F18 F19 0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181The Fibonacci sequence can be extended to negative integer indices by following the same recurrence relation in the negative direction (sequence A039834 in the OEIS): F 1 = 1 {\displaystyle F_{1}=1} , F 0 = 0 {\displaystyle F_{0}=0} , and F n = F n + 2 − F n + 1 {\displaystyle F_{n}=F_{n+2}-F_{n+1}} for n < 0 . Nearly all properties of Fibonacci numbers do not depend upon whether the indices are positive or negative. The values for positive and negative indices obey the relation:[10] F − n = ( − 1 ) n + 1 F n . {\displaystyle F_{-n}=(-1)^{n+1}F_{n}.}
History edit India edit See also: Golden ratio § History Thirteen (F7) ways of arranging long and short syllables in a cadence of length six. Eight (F6) end with a short syllable and five (F5) end with a long syllable.The Fibonacci sequence appears in Indian mathematics, in connection with Sanskrit prosody.[4][11][12] 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 Fm+1.[5]
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 (Fm+1) is obtained by adding one [S] to the Fm cases and one [L] to the Fm−1 cases.[13] Bharata Muni also expresses knowledge of the sequence in the Natya Shastra (c. 100 BC–c. 350 AD).[3][4] However, the clearest exposition of the sequence arises in the work of Virahanka (c. 700 AD), whose own work is lost, but is ailable in a quotation by Gopala (c. 1135):[12]
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,[3] writing that "the sum of the last and the one before the last is the number ... of the next mātrā-vṛtta."[15][16]
Europe edit A page of Fibonacci's Liber Abaci from the Biblioteca Nazionale di Firenze showing (in box on right) 13 entries of the Fibonacci sequence: the indices from present to XII (months) as Latin ordinals and Roman numerals and the numbers (of rabbit pairs) as Hindu-Arabic numerals starting with 1, 2, 3, 5 and ending with 377.The Fibonacci sequence first appears in the book Liber Abaci (The Book of Calculation, 1202) by Fibonacci,[17][18] where it is used to calculate the growth of rabbit populations.[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 rabbit math problem: 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 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]
Solution to Fibonacci rabbit problem: In a growing idealized population, the number of rabbit pairs form the Fibonacci sequence. At the end of the nth month, the number of pairs is equal to Fn. Relation to the golden ratio edit See also: Golden ratio and Golden field Closed-form expression editLike every sequence defined by a homogeneous linear recurrence with constant coefficients, the Fibonacci numbers he a closed-form expression.[22] 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]
F n = φ n − ψ n φ − ψ = φ n − ψ n 5 , {\displaystyle F_{n}={\frac {\varphi ^{n}-\psi ^{n}}{\varphi -\psi }}={\frac {\varphi ^{n}-\psi ^{n}}{\sqrt {5}}},}
where φ {\displaystyle \varphi } is the golden ratio and ψ {\displaystyle \psi } is its conjugate,[24]
φ = 1 2 ( 1 + 5 ) = − 1.61803 … , ψ = 1 2 ( 1 − 5 ) = − 0.61803 … . {\displaystyle {\begin{aligned}\varphi &={\tfrac {1}{2}}{\bigl (}1+{\sqrt {5}}~\!{\bigr )}={\phantom {-}}1.61803\ldots ,\\[5mu]\psi &={\tfrac {1}{2}}{\bigl (}1-{\sqrt {5}}~\!{\bigr )}=-0.61803\ldots .\end{aligned}}} The numbers φ {\displaystyle \varphi } and ψ {\displaystyle \psi } are the two solutions of the quadratic equation x 2 − x − 1 = 0 {\displaystyle \textstyle x^{2}-x-1=0} , that is, ( x − φ ) ( x − ψ ) = x 2 − x − 1 {\displaystyle (x-\varphi )(x-\psi )=x^{2}-x-1} , and thus they satisfy the identities φ + ψ = 1 {\displaystyle \varphi +\psi =1} and φ ψ = − 1 {\displaystyle \varphi \psi =-1} .
Since ψ = − φ − 1 {\displaystyle \psi =-\varphi ^{-1}} , Binet's formula can also be written as
F n = φ n − ( − φ ) − n 5 = φ n − ( − φ ) − n 2 φ − 1 . {\displaystyle F_{n}={\frac {\varphi ^{n}-(-\varphi )^{-n}}{\sqrt {5}}}={\frac {\varphi ^{n}-(-\varphi )^{-n}}{2\varphi -1}}.}
To see the relation between the sequence and these constants,[25] note that φ {\displaystyle \varphi } and ψ {\displaystyle \psi } are and thus x n = x n − 1 + x n − 2 , {\displaystyle x^{n}=x^{n-1}+x^{n-2},} so the powers of φ {\displaystyle \varphi } and ψ {\displaystyle \psi } satisfy the Fibonacci recurrence. In other words,
φ n = φ n − 1 + φ n − 2 , ψ n = ψ n − 1 + ψ n − 2 . {\displaystyle {\begin{aligned}\varphi ^{n}&=\varphi ^{n-1}+\varphi ^{n-2},\\[3mu]\psi ^{n}&=\psi ^{n-1}+\psi ^{n-2}.\end{aligned}}}
It follows that for any values a and b, the sequence defined by
U n = a φ n + b ψ n {\displaystyle U_{n}=a\varphi ^{n}+b\psi ^{n}}
satisfies the same recurrence,
U n = a φ n + b ψ n = a ( φ n − 1 + φ n − 2 ) + b ( ψ n − 1 + ψ n − 2 ) = a φ n − 1 + b ψ n − 1 + a φ n − 2 + b ψ n − 2 = U n − 1 + U n − 2 . {\displaystyle {\begin{aligned}U_{n}&=a\varphi ^{n}+b\psi ^{n}\\[3mu]&=a(\varphi ^{n-1}+\varphi ^{n-2})+b(\psi ^{n-1}+\psi ^{n-2})\\[3mu]&=a\varphi ^{n-1}+b\psi ^{n-1}+a\varphi ^{n-2}+b\psi ^{n-2}\\[3mu]&=U_{n-1}+U_{n-2}.\end{aligned}}}
If a and b are chosen so that U0 = 0 and U1 = 1 then the resulting sequence Un must be the Fibonacci sequence. This is the same as requiring a and b satisfy the system of equations:
{ a + b = 0 φ a + ψ b = 1 {\displaystyle \left\{{\begin{aligned}a+b&=0\\\varphi a+\psi b&=1\end{aligned}}\right.}
which has solution
a = 1 φ − ψ = 1 5 , b = − a , {\displaystyle a={\frac {1}{\varphi -\psi }}={\frac {1}{\sqrt {5}}},\quad b=-a,}
producing the required formula.
Taking the starting values U0 and U1 to be arbitrary constants, a more general solution is:
U n = a φ n + b ψ n {\displaystyle U_{n}=a\varphi ^{n}+b\psi ^{n}}
where
a = U 1 − U 0 ψ 5 , b = U 0 φ − U 1 5 . {\displaystyle {\begin{aligned}a&={\frac {U_{1}-U_{0}\psi }{\sqrt {5}}},\\[3mu]b&={\frac {U_{0}\varphi -U_{1}}{\sqrt {5}}}.\end{aligned}}}
Computation by rounding editSince | ψ n 5 |