Fibonacci Sequence Formula

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1.Fibonacci sequence history

Fibonacci number

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"Fibonacci Sequence" redirects here. For the chamber ensemble, see Fibonacci Sequence (ensemble).

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 Fn, 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,

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

and

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

for n > 1.

The beginning of the sequence is thus:

{\displaystyle 0,\;1,\;1,\;2,\;3,\;5,\;8,\;13,\;21,\;34,\;55,\;89,\;144,\;\ldots]

In some older books, 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.



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.

2.Fibonacci sequence number

Fibonacci numbers are named after Italian mathematician Leonardo of Pisa, later known as Fibonacci. In his 1202 book Liber Abaci, Fibonacci introduced the sequence to Western European mathematics,although the sequence had been described earlier 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.

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 sequence

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}}

Contents

1History

2Applications

2.1Music

2.2Nature

3Mathematics

3.1Sequence properties

3.2Relation to the golden ratio

3.2.1Closed-form expression

3.2.2Computation by rounding

3.2.3Limit of consecutive quotients

3.2.4Decomposition of powers

3.3Matrix form

3.4Identification

3.5Combinatorial identities

3.5.1Symbolic method

3.6Other identities

3.6.1Cassini's and Catalan's identities

3.6.2d'Ocagne's identity

3.7Power series

3.8Reciprocal sums

3.9Primes and divisibility

3.9.1Divisibility properties

3.9.2Primality testing

3.9.3Fibonacci primes

3.9.4Prime divisors

3.9.5Periodicity modulo n

3.10Right triangles

3.11Magnitude

3.12Generalizations

4See also

5References

5.1Works cited

6External links

History[edit]

See also: Golden ratio § History

3. How Fibonacci sequence works

Thirteen (F7) ways of arranging long (shown by the red tiles) and short syllables (shown by the grey squares) in a cadence of length six. Five (F5) end with a long syllable and eight (F6) end with a short syllable.

The Fibonacci sequence appears in Indian mathematics in connection with Sanskrit prosody, as pointed out by Parmanand Singh in 1985.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.

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 casesBharata Muni also expresses knowledge of the sequence in the Natya Shastra (c. 100 BC–c. 350 AD).[12][6] 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):

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].

Hemachandra (c. 1150) is credited with knowledge of the sequence as well,[6] writing that "the sum of the last and the one before the last is the number ... of the next mātrā-vṛtta."

4. Fibonacci sequence formula

A page of Fibonacci's Liber Abaci from the Biblioteca Nazionale di Firenze showing (in box on right) the Fibonacci sequence with the position in the sequence labeled in Latin and Roman numerals and the value in Hindu-Arabic numerals.

The number of rabbit pairs form the Fibonacci sequence

Outside India, the Fibonacci sequence first appears in the book Liber Abaci (1202) by Fibonacci.using it to calculate the growth of rabbit populations.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.

Fibonacci sequence

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.

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

Applications[edit]

The Fibonacci numbers are important in the 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.

Brasch et al. 2012 show how a generalised Fibonacci sequence also can be connected to the field of economics.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.

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.

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.

5. Fibonacci sequence golden ratio

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.

Fibonacci numbers are used by some pseudorandom number generators.

They are also used in planning poker, which is a step in estimating in software development projects that use the Scrum methodology.

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.

Fibonacci numbers arise in the analysis of the Fibonacci heap data structure.

The Fibonacci cube is an undirected graph with a Fibonacci number of nodes that has been proposed as a network topology for parallel computing.

A one-dimensional optimization method, called the Fibonacci search technique, uss Fibonacci numbers.

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.

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.

Fibonacci sequence

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.)

Mario Merz included the Fibonacci sequence in some of his works beginning in 1970.

Fibonacci retracement levels are widely used in technical analysis for financial market trading.

Fibonacci numbers appear in the ring lemma, used to prove connections between the circle packing theorem and conformal maps.

Music[edit]

See also: Golden ratio § Music

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.

Nature[edit]

Further information: Patterns in nature

See also: Golden ratio § Nature

7. Fibonacci sequence Python

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,such as branching in trees, arrangement of leaves on a stem, the fruitlets of a pineapple, the flowering of artichoke, an uncurling fern and the arrangement of a pine cone, and the family tree of honeybee][36] Kepler pointed out the presence of the Fibonacci sequence in nature, using it to explain the (golden ratio-related) pentagonal form of some flowers.Field daisies most often have petals in counts of Fibonacci numbers. In 1754, Charles Bonnet discovered that the spiral phyllotaxis of plants were frequently expressed in Fibonacci number series.

Fibonacci sequence tools

Przemysław Prusinkiewicz advanced the idea that real instances can in part be understood a by Helmut Vogel [de] in 1979.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,typically counted by the outermost range of radii.

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, Fn, is the number of female ancestors, which is Fn−1, plus the number of male ancestors, which is Fn−2.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".])

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.[45] 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.

Mathematics

Fibonacci sequence

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):

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

These numbers also give the solution to certain enumerative problems,[48] 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 Fn+1 ways to do this. For example, if n = 5, then Fn+1 = F6 = 8 counts the eight compositions summing to 5:

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

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 Fn+2. For example, out of the 16 binary strings of length 4, there are F6 = 8 without consecutive 1s – they are 0000, 0001, 0010, 0100, 0101, 1000, 1001, and 1010. Equivalently, Fn+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.

The number of binary strings of length n without an odd number of consecutive 1s is the Fibonacci number Fn+1. For example, out of the 16 binary strings of length 4, there are F5 = 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 Fn+1.

The number of binary strings of length n without an even number of consecutive 0s or 1s is 2Fn. For example, out of the 16 binary strings of length 4, there are 2F4 = 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.

Sequence properties[edit]

The first 21 Fibonacci numbers Fn are:

F0F1F2F3F4F5F6F7F8F9F10F11F12F13F14F15F16F17F18F19F20011235813213455891442333776109871597258441816765

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 satisfying

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

Thus the bidirectional sequence is

F−8F−7F−6F−5F−4F−3F−2F−1F0F1F2F3F4F5F6F7F8−2113−85−32−1101123581321

Relation to the golden ratio[edit]

Main article: Golden ratio

Closed-form expression[edit]

Like every sequence defined by a linear recurrence with constant coefficients, the Fibonacci numbers have a closed-form solution. It has become known as "Binet's formula", though it was already known by Abraham de Moivre and Daniel Bernoulli:

{\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

{\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,[52] 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}=a\varphi ^{n-1}+b\psi ^{n-1}+a\varphi ^{n-2}+b\psi ^{n-2}=U_{n-1}+U_{n-2}.}

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:

{\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 U0 and U1 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 Fn is the closest integer to {\displaystyle {\frac {\varphi ^{n}}{\s 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 }

{\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 Fn 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 An Am = An+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{aligned}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}.\end{aligned}}}

Fibonacci sequence equation

What is the Fibonacci sequence

Why Fibonacci sequence for story points

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).

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.[56] This is because Binet's formula above can be rearranged to give

{\displaystyle n=\log _{\varphi }\left({\frac {F_{n}{\sqrt {5}}+{\sqrt {5F_{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

Most identities involving Fibonacci numbers can be proved using combinatorial arguments using the fact that Fn can be interpreted as the number of sequences of 1s and 2s that sum to n − 1. This can be taken as the definition of Fn, with the convention that F0 = 0, meaning no sum adds up to −1, and that F1 = 1, meaning the empty sum "adds up" to 0. Here, the order of the summand matters. For example, 1 + 2 and 2 + 1 are considered two different sums.

For example, the recurrence relation

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

or in words, the nth Fibonacci number is the sum of the previous two Fibonacci numbers, may be shown by dividing the Fn sums of 1s and 2s that add to n − 1 into two non-overlapping groups. One group contains those sums whose first term is 1 and the other those sums whose first term is 2. In the first group the remaining terms add to n − 2, so it has Fn-1 sums, and in the second group the remaining terms add to n − 3, so there are Fn−2 sums. So there are a total of Fn−1 + Fn−2 sums altogether, showing this is equal to Fn.

Similarly, 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.[57] In symbols:

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

This is done by dividing the sums adding to n + 1 in a different way, this time by the location of the first 2. Specifically, the first group consists of those sums that start with 2, the second group those that start 1 + 2, the third 1 + 1 + 2, and so on, until the last group, which consists of the single sum where only 1's are used. The number of sums in the first group is F(n), F(n − 1) in the second group, and so on, with 1 sum in the last group. So the total number of sums is F(n) + F(n − 1) + ... + F(1) + 1 and therefore this quantity is equal to F(n + 2).

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 F2n−1 is the (2n)th Fibonacci number, and the sum of the first Fibonacci numbers with even index up to F2n is the (2n + 1)th Fibonacci number minus 1.[58]

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 Fn is the product of the nth and (n + 1)th Fibonacci numbers. In this case Fibonacci rectangle of size Fn by F(n + 1) can be decomposed into squares of size Fn, Fn−1, and so on to F1 = 1, from which 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.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.

The Fibonacci Sequence is the series of numbers:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...

The next number is found by adding up the two numbers before it:

the 2 is found by adding the two numbers before it (1+1),

the 3 is found by adding the two numbers before it (1+2),

the 5 is (2+3),

and so on!

Example: the next number in the sequence above is 21+34 = 55

It is that simple!

Here is a longer list:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, ...

Can you figure out the next few numbers?

Makes A Spiral

When we make squares with those widths, we get a nice spiral:

Do you see how the squares fit neatly together?

For example 5 and 8 make 13, 8 and 13 make 21, and so on.

This spiral is found in nature!

See: Nature, The Golden Ratio, and Fibonacci

The Rule

The Fibonacci Sequence can be written as a "Rule" (see Sequences and Series).

First, the terms are numbered from 0 onwards like this:

n =01234567891011121314...xn =01123581321345589144233377...

So term number 6 is called x6 (which equals 8).

Example: the 8th term is

the 7th term plus the 6th term:

x8 = x7 + x6



So we can write the rule:

The Rule is xn = xn−1 + xn−2

where:

xn is term number "n"

xn−1 is the previous term (n−1)

xn−2 is the term before that (n−2)

Example: term 9 is calculated like this:

x9= x9−1 + x9−2

 = x8 + x7

 = 21 + 13

 = 34

Golden Ratio



And here is a surprise. When we take any two successive (one after the other) Fibonacci Numbers, their ratio is very close to the Golden Ratio "φ" which is approximately 1.618034...

In fact, the bigger the pair of Fibonacci Numbers, the closer the approximation. Let us try a few:

A

B

 

B / A

2

3

 1.5

3

5

 1.666666666...

5

8

 1.6

8

13

 1.625

...

...

 ...

144

233

 1.618055556...

233

377

 1.618025751...

...

...

 ...

We don't have to start with 2 and 3, here I randomly chose 192 and 16 (and got the sequence 192, 16, 208, 224, 432, 656, 1088, 1744, 2832, 4576, 7408, 11984, 19392, 31376, ...):

A

B

 

B / A

192

16

 0.08333333...

16

208

 13

208

224

 1.07692308...

224

432

 1.92857143...

...

...

 ...

7408

11984

 1.61771058...

11984

19392

 1.61815754...

...

...

 ...

It takes longer to get good values, but it shows that not just the Fibonacci Sequence can do this!

Using The Golden Ratio to Calculate Fibonacci Numbers

And even more surprising is that we can calculate any Fibonacci Number using the Golden Ratio:

xn = φn − (1−φ)n√5

The answer comes out as a whole number, exactly equal to the addition of the previous two terms.

Example: x6

x6 = (1.618034...)6 − (1−1.618034...)6√5

When I used a calculator on this (only entering the Golden Ratio to 6 decimal places) I got the answer 8.00000033 , a more accurate calculation would be closer to 8.

Try n=12 and see what you get.

You can also calculate a Fibonacci Number by multiplying the previous Fibonacci Number by the Golden Ratio and then rounding (works for numbers above 1):

Example: 8 × φ = 8 × 1.618034... = 12.94427... = 13 (rounded)

Some Interesting Things

Here is the Fibonacci sequence again:

n =0123456789101112131415...xn =01123581321345589144233377610...

There is an interesting pattern:

Look at the number x3 = 2. Every 3rd number is a multiple of 2 (2, 8, 34, 144, 610, ...)

Look at the number x4 = 3. Every 4th number is a multiple of 3 (3, 21, 144, ...)

Look at the number x5 = 5. Every 5th number is a multiple of 5 (5, 55, 610, ...)

And so on (every nth number is a multiple of xn).

 

1/89 = 0.011235955056179775...

Notice the first few digits (0,1,1,2,3,5) are the Fibonacci sequence?

In a way they all are, except multiple digit numbers (13, 21, etc) overlap, like this:

0.00.010.0010.00020.000030.0000050.00000080.000000130.000000021    ... etc ...

0.011235955056179775...  =  1/89

 

Terms Below Zero

The sequence works below zero also, like this:

n =...−6−5−4−3−2−10123456...xn =...−85−32−110112358...

(Prove to yourself that each number is found by adding up the two numbers before it!)

In fact the sequence below zero has the same numbers as the sequence above zero, except they follow a +-+- ... pattern. It can be written like this:

x−n = (−1)n+1 xn

Which says that term "−n" is equal to (−1)n+1 times term "n", and the value (−1)n+1 neatly makes the correct +1, −1, +1, −1, ... pattern.

History

Fibonacci was not the first to know about the sequence, it was known in India hundreds of years before!



About Fibonacci The Man

His real name was Leonardo Pisano Bogollo, and he lived between 1170 and 1250 in Italy.

"Fibonacci" was his nickname, which roughly means "Son of Bonacci".

As well as being famous for the Fibonacci Sequence, he helped spread Hindu-Arabic Numerals (like our present numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9) through Europe in place of Roman Numerals (I, II, III, IV, V, etc). That has saved us all a lot of trouble! Thank you Leonardo.

Fibonacci Day

Fibonacci Day is November 23rd, as it has the digits "1, 1, 2, 3" which is part of the sequence. So next Nov 23 let everyone know!