
In mathematics, the Fibonacci numbers, commonly denoted , form a
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...
, the Fibonacci sequence, in which each number is the sum of the two preceding ones. 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 first few values in the sequence are:
:0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144.
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. They are named after the Italian mathematician Leonardo of Pisa, later known as Fibonacci, who introduced the sequence to Western European mathematics in his 1202 book '' Liber Abaci''.
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
The globe artichoke ('' Cynara cardunculus'' var. ''scolymus'' ),Rottenberg, A., and D. Zohary, 1996: "The wild ancestry of the cultivated artichoke." Genet. Res. Crop Evol. 43, 53–58. also known by the names French artichoke and green artich ...
, an uncurling fern
A fern (Polypodiopsida or Polypodiophyta ) is a member of a group of vascular plants (plants with xylem and phloem) that reproduce via spores and have neither seeds nor flowers. The polypodiophytes include all living pteridophytes except ...
, and the arrangement of a pine cone's bracts.
Fibonacci numbers are also strongly related to the golden ratio
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0,
where the Greek letter phi ( ...
: Binet's formula expresses the th Fibonacci number in terms of and the golden ratio, and implies that the ratio of two consecutive Fibonacci numbers tends to the golden ratio as 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.
Definition
The Fibonacci numbers may be defined by the recurrence relation
and
for .
Under some older definitions, the value is omitted, so that the sequence starts with and the recurrence is valid for .
The first 20 Fibonacci numbers are:[
:
]
History
The Fibonacci sequence appears in Indian mathematics, in connection with Sanskrit prosody. 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 units is .
Knowledge of the Fibonacci sequence was expressed as early as Pingala ( 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 beats () is obtained by adding one to the cases and one to the cases. Bharata Muni also expresses knowledge of the sequence in the '' Natya Shastra'' (c. 100 BC–c. 350 AD).
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 s the variation
S, or s, is the nineteenth letter in the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''ess'' (pronounced ), plural ''esses''.
History ...
.. For example, for meter of lengthfour, variations of meters of two ndthree being mixed, five happens. orks out examples 8, 13, 21
Ork or ORK may refer to:
* Ork (folklore), a mountain demon of Tyrol folklore
* ''Ork'' (video game), a 1991 game for the Amiga and Atari ST systems
* Ork (''Warhammer 40,000''), a fictional species in the ''Warhammer 40,000'' universe
* ''Ork!'' ...
.. In this way, the process should be followed in all ''mātrā-vṛttas'' rosodic combinations
Hemachandra (c. 1150) is credited with knowledge of the sequence as well,[ writing that "the sum of the last and the one before the last is the number ... of the next mātrā-vṛtta."
Outside India, the Fibonacci sequence first appears in the book '' Liber Abaci'' (''The Book of Calculation'', 1202) by Fibonacci where it is used to calculate the growth of rabbit populations. Fibonacci considers the growth of an idealized (biologically unrealistic) ]rabbit
Rabbits, also known as bunnies or bunny rabbits, are small mammals in the family Leporidae (which also contains the hares) of the order Lagomorpha (which also contains the pikas). ''Oryctolagus cuniculus'' includes the European rabbit s ...
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 th month, the number of pairs of rabbits is equal to the number of mature pairs (that is, the number of pairs in month ) plus the number of pairs alive last month (month ). The number in the th month is the th Fibonacci number.
The name "Fibonacci sequence" was first used by the 19th-century number theorist Édouard Lucas.
Relation to the golden ratio
Closed-form expression
Like every sequence defined by a linear recurrence with constant coefficients, the Fibonacci numbers have a closed-form expression
In mathematics, a closed-form expression is a mathematical expression that uses a finite number of standard operations. It may contain constants, variables, certain well-known operations (e.g., + − × ÷), and functions (e.g., ''n''th ro ...
. 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
Daniel Bernoulli FRS (; – 27 March 1782) was a Swiss mathematician and physicist and was one of the many prominent mathematicians in the Bernoulli family from Basel. He is particularly remembered for his applications of mathematics to mech ...
:
where
is the golden ratio
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0,
where the Greek letter phi ( ...
, and is its conjugate:
Since , this formula can also be written as
To see the relation between the sequence and these constants, note that and are both solutions of the equation
so the powers of and satisfy the Fibonacci recursion. In other words,
and
It follows that for any values and , the sequence defined by
satisfies the same recurrence.
If and are chosen so that and then the resulting sequence must be the Fibonacci sequence. This is the same as requiring and satisfy the system of equations:
which has solution
producing the required formula.
Taking the starting values and to be arbitrary constants, a more general solution is:
where
Computation by rounding
Since
for all , the number is the closest integer to . Therefore, it can be found by rounding
Rounding means replacing a number with an approximate value that has a shorter, simpler, or more explicit representation. For example, replacing $ with $, the fraction 312/937 with 1/3, or the expression with .
Rounding is often done to obta ...
, using the nearest integer function:
In fact, the rounding error is very small, being less than 0.1 for , and less than 0.01 for .
Fibonacci numbers can also be computed by truncation, in terms of the floor function:
As the floor function is monotonic, the latter formula can be inverted for finding the index of the smallest Fibonacci number that is not less than a positive integer :
where , , and .
Magnitude
Since ''Fn'' is asymptotic to , the number of digits in ''F''''n'' is asymptotic to . 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
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
This convergence holds regardless of the starting values and , unless . 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, , because the ratios between consecutive Fibonacci numbers approaches .
:
Decomposition of powers
Since the golden ratio satisfies the equation
this expression can be used to decompose higher powers as a linear function of lower powers, which in turn can be decomposed all the way down to a linear combination of and 1. The resulting recurrence relationships yield Fibonacci numbers as the linear coefficients:
This equation can be proved by induction on :
For , it is also the case that and it is also the case that
These expressions are also true for if the Fibonacci sequence ''Fn'' is extended to negative integers using the Fibonacci rule
Identification
Binet's formula provides a proof that a positive integer ''x'' is a Fibonacci number if and only if at least one of or is a perfect square
''Perfect Square'' is a 2004 concert film of the alternative rock Musical ensemble, band R.E.M. (band), R.E.M., filmed on July 19, 2003, at the bowling green, Bowling Green in Wiesbaden, Germany. It was released by Warner Reprise Video on March 9, ...
. This is because Binet's formula, which can be written as , can be multiplied by and solved as a quadratic equation in via the quadratic formula:
Comparing this to , it follows that
:
In particular, the left-hand side is a perfect square.
Matrix form
A 2-dimensional system of linear difference equations that describes the Fibonacci sequence is
alternatively denoted
which yields . The eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denot ...
s of the matrix are and corresponding to the respective eigenvector
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...
s
and
As the initial value is
it follows that the th term is
From this, the th element in the Fibonacci series
may be read off directly as a closed-form expression
In mathematics, a closed-form expression is a mathematical expression that uses a finite number of standard operations. It may contain constants, variables, certain well-known operations (e.g., + − × ÷), and functions (e.g., ''n''th ro ...
:
Equivalently, the same computation may performed by diagonalization of through use of its eigendecomposition:
where and
The closed-form expression for the th element in the Fibonacci series is therefore given by
which again yields
The matrix has a determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if ...
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:
The Fibonacci numbers occur as the ratio of successive convergents of the continued fraction for , 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 , this matrix can be computed in arithmetic operations, using the exponentiation by squaring method.
Taking the determinant of both sides of this equation yields Cassini's identity,
Moreover, since for any square matrix , 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 into ),
In particular, with ,
These last two identities provide a way to compute Fibonacci numbers recursively in arithmetic operations and in time , where is the time for the multiplication of two numbers of digits. This matches the time for computing the 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
In computing, memoization or memoisation is an optimization technique used primarily to speed up computer programs by storing the results of expensive function calls and returning the cached result when the same inputs occur again. Memoization ...
).
Combinatorial identities
Combinatorial proofs
Most identities involving Fibonacci numbers can be proved using combinatorial arguments using the fact that can be interpreted as the number of ossibly emptysequences of 1s and 2s whose sum is . This can be taken as the definition of with the conventions , meaning no such sequence exists whose sum is −1, and , meaning the empty sequence "adds up" to 0. In the following, is the cardinality
In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
of a set:
:
:
:
:
:
:
In this manner the recurrence relation
may be understood by dividing the sequences into two non-overlapping sets where all sequences either begin with 1 or 2:
Excluding the first element, the remaining terms in each sequence sum to or and the cardinality
In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
of each set is or giving a total of sequences, showing this is equal to .
In a similar manner it may be shown that the sum of the first Fibonacci numbers up to the ''n''th is equal to the (''n'' + 2)-nd Fibonacci number minus 1. In symbols:
This may be seen by dividing all sequences summing to based on the location of the first 2. Specifically, each set consists of those sequences that start until the last two sets 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 . From this it follows that .
A similar argument, grouping the sums by the position of the first 1 rather than the first 2 gives two more identities:
and
In words, the sum of the first Fibonacci numbers with odd index up to is the (2''n'')th Fibonacci number, and the sum of the first Fibonacci numbers with even index up to is the (2''n'' + 1)th Fibonacci number minus 1.
A different trick may be used to prove
or in words, the sum of the squares of the first Fibonacci numbers up to is the product of the ''n''th and (''n'' + 1)th Fibonacci numbers. To see this, begin with a Fibonacci rectangle of size and decompose it into squares of size ; from this the identity follows by comparing areas:
Symbolic method
The sequence is also considered using the symbolic method. More precisely, this sequence corresponds to a specifiable combinatorial class
In combinatorics, the symbolic method is a technique for counting combinatorial objects. It uses the internal structure of the objects to derive formulas for their generating functions. The method is mostly associated with Philippe Flajolet and ...
. The specification of this sequence is . Indeed, as stated above, the -th Fibonacci number equals the number of combinatorial compositions (ordered partitions) of using terms 1 and 2.
It follows that the ordinary generating function of the Fibonacci sequence, i.e. , is the complex function .
Induction proofs
Fibonacci identities often can be easily proved using mathematical induction
Mathematical induction is a method for proving that a statement ''P''(''n'') is true for every natural number ''n'', that is, that the infinitely many cases ''P''(0), ''P''(1), ''P''(2), ''P''(3), ... all hold. Informal metaphors help ...
.
For example, reconsider
Adding to both sides gives
:
and so we have the formula for
Similarly, add to both sides of
to give
Binet formula proofs
The Binet formula is
This can be used to prove Fibonacci identities.
For example, to prove that
note that the left hand side multiplied by becomes
as required, using the facts and to simplify the equations.
Other identities
Numerous other identities can be derived using various methods. Here are some of them:
Cassini's and Catalan's identities
Cassini's identity states that
Catalan's identity is a generalization:
d'Ocagne's identity
where ''L''''n'' is the ''nth Lucas number. The last is an identity for doubling ''n''; other identities of this type are
by Cassini's identity.
These can be found experimentally using lattice reduction, and are useful in setting up the special number field sieve to factorize
In mathematics, factorization (or factorisation, see English spelling differences) or factoring consists of writing a number or another mathematical object as a product of several ''factors'', usually smaller or simpler objects of the same kin ...
a Fibonacci number.
More generally,[
or alternatively
Putting in this formula, one gets again the formulas of the end of above section Matrix form.
]
Generating function
The generating function
In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary ser ...
of the Fibonacci sequence is the power series
This series is convergent for and its sum has a simple closed-form:
This can be proved by using the Fibonacci recurrence to expand each coefficient in the infinite sum:
Solving the equation
for ''s''(''x'') results in the closed form.
The partial fraction decomposition is given by
where is the golden ratio and is its conjugate.
gives the generating function for the negafibonacci numbers, and satisfies the functional equation
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
and the sum of squared reciprocal Fibonacci numbers as
and there is a ''nested'' sum of squared Fibonacci numbers giving the reciprocal of the golden ratio
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0,
where the Greek letter phi ( ...
,
The sum of all even-indexed reciprocal Fibonacci numbers is
with the Lambert series since
So the reciprocal Fibonacci constant is
Moreover, this number has been proved irrational by Richard André-Jeannin.
Millin's series gives the identity
which follows from the closed form for its partial sums as ''N'' tends to infinity:
Primes and divisibility
Divisibility properties
Every third number of the sequence is even (a multiple of ) and, more generally, every ''k''th number of the sequence is a multiple of ''Fk''. Thus the Fibonacci sequence is an example of a divisibility sequence. In fact, the Fibonacci sequence satisfies the stronger divisibility property
where is the greatest common divisor function.
In particular, any three consecutive Fibonacci numbers are pairwise coprime because both and . That is,
:
for every ''n''.
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.
These cases can be combined into a single, non-piecewise
In mathematics, a piecewise-defined function (also called a piecewise function, a hybrid function, or definition by cases) is a function defined by multiple sub-functions, where each sub-function applies to a different interval in the domain. P ...
formula, using the Legendre symbol:
Primality testing
The above formula can be used as a primality test in the sense that if
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 largesay a 500-bit numberthen we can calculate ''F''''m'' (mod ''n'') efficiently using the matrix form. Thus
Here the matrix power ''A''''m'' is calculated using modular exponentiation, which can be adapted to matrices.
Fibonacci primes
A ''Fibonacci prime'' is a Fibonacci number that is prime
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only way ...
. The first few are:
: 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.
''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 number
A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Equivalently, it is a positive integer that has at least one divisor other than 1 and itself. Every positive integer is composite, prime, ...
s, 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.
The only nontrivial square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
Fibonacci number is 144. Attila Pethő proved in 2001 that there is only a finite number of perfect power Fibonacci numbers. In 2006, Y. Bugeaud, M. Mignotte, and S. Siksek proved that 8 and 144 are the only such non-trivial perfect powers.
1, 3, 21, and 55 are the only triangular Fibonacci numbers, which was conjectured by Vern Hoggatt and proved by Luo Ming.
No Fibonacci number can be a perfect number
In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number.
T ...
. More generally, no Fibonacci number other than 1 can be multiply perfect, and no ratio of two Fibonacci numbers can be perfect.
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). As a result, 8 and 144 (''F''6 and ''F''12) are the only Fibonacci numbers that are the product of other Fibonacci numbers.
The divisibility of Fibonacci numbers by a prime ''p'' is related to the Legendre symbol which is evaluated as follows:
If ''p'' is a prime number then
For example,
It is not known whether there exists a prime ''p'' such that
Such primes (if there are any) would be called Wall–Sun–Sun primes.
Also, if ''p'' ≠ 5 is an odd prime number then:
Example 1. ''p'' = 7, in this case ''p'' ≡ 3 (mod 4) and we have:
Example 2. ''p'' = 11, in this case ''p'' ≡ 3 (mod 4) and we have:
Example 3. ''p'' = 13, in this case ''p'' ≡ 1 (mod 4) and we have:
Example 4. ''p'' = 29, in this case ''p'' ≡ 1 (mod 4) and we have:
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.
For example,
All known factors of Fibonacci numbers ''F''(''i'') for all ''i'' < 50000 are collected at the relevant repositories.
Periodicity modulo ''n''
If the members of the Fibonacci sequence are taken mod ''n'', the resulting sequence is periodic with period at most ''6n''. The lengths of the periods for various ''n'' form the so-called Pisano periods. 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
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subt ...
. However, for any particular ''n'', the Pisano period may be found as an instance of cycle detection.
Generalizations
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.[
* Starting with other integers. Lucas numbers have ''L''1 = 1, ''L''2 = 3, and ''Ln'' = ''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 ''Pn'' = 2''P''''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''.
]
Applications
Mathematics
The Fibonacci numbers occur in the sums of "shallow" diagonals in Pascal's triangle
In mathematics, Pascal's triangle is a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, althoug ...
(see binomial coefficient
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
):
The generating function can be expanded into
and collecting like terms of , we have the identity
To see how the formula is used, we can arrange the sums by the number of terms present:
:{,
,
,
, -
,
,
,
,
,
, -
,
,
,
,
which is , where we are choosing the positions of ''k'' twos from ''n-k-1'' terms.
These numbers also give the solution to certain enumerative problems, the most common of which is that of counting the number of ways of writing a given number as an ordered sum of 1s and 2s (called compositions); there are ways to do this (equivalently, it's also the number of domino tilings of the rectangle). For example, there are ways one can climb a staircase of 5 steps, taking one or two steps at a time:
:{,
,
,
,
,
,
,
, -
,
,
,
,
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 subset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...
s of a given set.
* The number of binary strings of length without consecutive s is the Fibonacci number . For example, out of the 16 binary strings of length 4, there are without consecutive s – they are 0000, 0001, 0010, 0100, 0101, 1000, 1001, and 1010. Such strings are the binary representations of Fibbinary numbers. Equivalently, is the number of subsets of without consecutive integers, that is, those for which for every . 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 without an odd number of consecutive s is the Fibonacci number . For example, out of the 16 binary strings of length 4, there are without an odd number of consecutive s – they are 0000, 0011, 0110, 1100, 1111. Equivalently, the number of subsets of without an odd number of consecutive integers is . A bijection with the sums to ''n'' is to replace ''1'' with ''0'' and ''2'' with ''11''.
* The number of binary strings of length without an even number of consecutive s or s is . For example, out of the 16 binary strings of length 4, there are without an even number of consecutive s or s – 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.
*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.
*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
In the geometry of circle packings in the Euclidean plane, the ring lemma gives a lower bound on the sizes of adjacent circles in a circle packing.
Statement
The lemma states: Let n be any integer greater than or equal to three. Suppose that the ...
, used to prove connections between the circle packing theorem and conformal map
In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths.
More formally, let U and V be open subsets of \mathbb^n. A function f:U\to V is called conformal (or angle-preserving) at a point u_0\in ...
s.
Computer science
*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.
*Fibonacci numbers are used in a polyphase version of the merge sort
In computer science, merge sort (also commonly spelled as mergesort) is an efficient, general-purpose, and comparison-based sorting algorithm. Most implementations produce a stable sort, which means that the order of equal elements is the same ...
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
A polyphase merge sort is a variation of a bottom-up merge sort that sorts a list using an initial uneven distribution of sub-lists (runs), primarily used for external sorting, and is more efficient than an ordinary merge sort when there are fewer ...
was described in '' The Art of Computer Programming''.
*A Fibonacci tree is a binary tree whose child trees (recursively) differ in height
Height is measure of vertical distance, either vertical extent (how "tall" something or someone is) or vertical position (how "high" a point is).
For example, "The height of that building is 50 m" or "The height of an airplane in-flight is ab ...
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.
*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.
*The Fibonacci number series is used for optional lossy compression in the IFF
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bicondi ...
8SVX audio file format used on Amiga computers. The number series compands the original audio wave similar to logarithmic methods such as μ-law.
*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.
* Fibonacci coding
* NegaFibonacci coding
Nature
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
The globe artichoke ('' Cynara cardunculus'' var. ''scolymus'' ),Rottenberg, A., and D. Zohary, 1996: "The wild ancestry of the cultivated artichoke." Genet. Res. Crop Evol. 43, 53–58. also known by the names French artichoke and green artich ...
, an uncurling fern and the arrangement of a pine cone, and the family tree of honeybees. Kepler pointed out the presence of the Fibonacci sequence in nature, using it to explain the (golden ratio
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0,
where the Greek letter phi ( ...
-related) pentagonal form of some flowers. Field daisies most often have petals in counts of Fibonacci numbers. In 1830, K. F. Schimper and A. Braun discovered that the parastichies (spiral phyllotaxis
In botany, phyllotaxis () or phyllotaxy is the arrangement of leaves on a plant stem. Phyllotactic spirals form a distinctive class of patterns in nature.
Leaf arrangement
The basic arrangements of leaves on a stem are opposite and alternat ...
) of plants were frequently expressed as fractions involving Fibonacci numbers.
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.
A model for the pattern of florets in the head of a sunflower was proposed by in 1979. This has the form
where is the index number of the floret and 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 , the nearest neighbors of floret number are those at for some index , which depends on , 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
A drone is a male honey bee. Unlike the female worker bee, drones do not have stingers. They gather neither nectar nor pollen and are unable to feed without assistance from worker bees. A drone's only role is to mate with a maiden queen in nupt ...
.
* 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, , is the number of female ancestors, which is , plus the number of male ancestors, which is . 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
The X chromosome is one of the two sex-determining chromosomes ( allosomes) in many organisms, including mammals (the other is the Y chromosome), and is found in both males and females. It is a part of the XY sex-determination system and XO se ...
inheritance line at a given ancestral generation also follows the Fibonacci sequence. A male individual has an X chromosome, which he received from his mother, and a Y chromosome
The Y chromosome is one of two sex chromosomes ( allosomes) in therian mammals, including humans, and many other animals. The other is the X chromosome. Y is normally the sex-determining chromosome in many species, since it is the presence or a ...
, which he received from his father. The male counts as the "origin" of his own X chromosome (), and at his parents' generation, his X chromosome came from a single parent (). 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 (). 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 (). Five great-great-grandparents contributed to the male descendant's X chromosome (), 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
Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultra ...
, 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 reflections, for , is the th Fibonacci number. (However, when , there are three reflection paths, not two, one for each of the three surfaces.)
* 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.
*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.
*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.
*Mario Merz
Mario Merz (1 January 1925 – 9 November 2003) was an Italian artist, and husband of Marisa Merz.
Life
Born in Milan, Merz started drawing during World War II, when he was imprisoned for his activities with the '' Giustizia e Libertà'' antif ...
included the Fibonacci sequence in some of his artworks beginning in 1970.
* 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. See also .
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External links
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Periods of Fibonacci Sequences Mod m
at MathPages
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Articles containing proofs