Presents, a Life with a Plan. My name is Karen Anastasia Placek, I am the author of this Google Blog. This is the story of my journey, a quest to understanding more than myself. The title of my first blog delivered more than a million views!! The title is its work as "The Secret of the Universe is Choice!; know decision" will be the next global slogan. Placed on T-shirts, Jackets, Sweatshirts, it really doesn't matter, 'cause a picture with my slogan is worth more than a thousand words, it's worth??.......Know Conversation!!!
Fibonacci Sequence Changes And The Pattern Of Repeating Numbers Is Present!!!!
Fibonacci sequence following the integer series is the first line however should logic be found than the in the second line I'll represent the last two numbers that would be added together breaking that sequence and creating a repeating pattern. The third line showing as represented, should the mathematics at the time represent a Roman Numeral as a number than the identification of just Pi (π) would be seen by designation as should the 'Parallel line' be sawn than the Fibonacci sequence would be the difference by the actual mathematics to an introduction of new mathematics(?).
A tiling with squares whose side lengths are successive Fibonacci numbers
In mathematics, the Fibonacci numbers are the numbers in the following integer sequence, called the Fibonacci sequence, and characterized by the fact that every number after the first two is the sum of the two preceding ones:[1][2]
Often, especially in modern usage, the sequence is extended by one more initial term:
The Fibonacci spiral: an approximation of the golden spiral created by drawing circular arcs connecting the opposite corners of squares in the Fibonacci tiling;[4] this one uses squares of sizes 1, 1, 2, 3, 5, 8, 13 and 21.
By definition, the first two numbers in the Fibonacci sequence are
either 1 and 1, or 0 and 1, depending on the chosen starting point of
the sequence, and each subsequent number is the sum of the previous two.
The sequence Fn of Fibonacci numbers is defined by the recurrence relation:
Fibonacci numbers appear to have first arisen in perhaps 200 BC in work by Pingala
on enumerating possible patterns of poetry formed from syllables of two
lengths. The Fibonacci sequence is named after Italian mathematician
Leonardo of Pisa, known as Fibonacci. His 1202 book Liber Abaci introduced the sequence to Western European mathematics,[6] although the sequence had been described earlier in Indian mathematics.[7][8][9] The sequence described in Liber Abaci began with F1 = 1. Fibonacci numbers were later independently discussed by Johannes Kepler
in 1611 in connection with approximations to the pentagon. Their
recurrence relation appears to have been understood from the early
1600s, but it has only been in the past very few decades that they have
in general become widely discussed.[10]
Fibonacci numbers are closely related to Lucas numbers in that they form a complementary pair of Lucas sequences and . They are intimately connected with the golden ratio; for example, the closest rational approximations to the ratio are 2/1, 3/2, 5/3, 8/5, ... .
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,[11] such as branching in trees, phyllotaxis (the arrangement of leaves on a stem), the fruit sprouts of a pineapple,[12] the flowering of an artichoke, an uncurling fern and the arrangement of a pine cone's bracts.[13]
Thirteen
ways of arranging long and short syllables in a cadence of length six.
Five end with a long syllable and eight end with a short syllable.
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 Fibonacci sequence appears in Indian mathematics, in connection with Sanskrit prosody.[8][14]
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.[9]
Susantha Goonatilake writes that the development of the Fibonacci sequence "is attributed in part to Pingala (200 BC), later being associated with Virahanka (c. 700 AD), Gopāla (c. 1135), and Hemachandra (c. 1150)".[7] Parmanand Singh cites Pingala's cryptic formula misrau cha ("the two are mixed") and cites 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. He dates Pingala before 450 BC.[15]
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].[16]
The number of rabbit pairs form the Fibonacci sequence
Outside India, the Fibonacci sequence first appears in the book Liber Abaci (1202) by Fibonacci.[6] Fibonacci considers the growth of a hypothetical, idealized (biologically unrealistic) rabbit
population, assuming that: a newly born pair of rabbits, one male, one
female, are put in a field; rabbits are able to mate at the age of one
month so that at the end of its second month a female can produce
another pair of rabbits; rabbits never die and a mating pair always
produces one new pair (one male, one female) every month from the second
month on. 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 the female produces a new pair, so now there are 2 pairs of rabbits in the field.
At the end of the third month, the original female produces a second pair, making 3 pairs in all in the field.
At the end of the fourth month, the original female has produced yet
another new pair, and the female born two months ago also produces her
first pair, making 5 pairs.
At the end of the nth month, the number of pairs of rabbits is equal to the number of new pairs (that is, the number of pairs in month n − 2) plus the number of pairs alive last month (that is, n − 1). This is the nth Fibonacci number.[17]
The name "Fibonacci sequence" was first used by the 19th-century number theorist Édouard Lucas.[18]
List of Fibonacci numbers
The first 21 Fibonacci numbers Fn for n = 0, 1, 2, …, 20 are:[19]
F0
F1
F2
F3
F4
F5
F6
F7
F8
F9
F10
F11
F12
F13
F14
F15
F16
F17
F18
F19
F20
0
1
1
2
3
5
8
13
21
34
55
89
144
233
377
610
987
1597
2584
4181
6765
The sequence can also be extended to negative index n using the re-arranged recurrence relation
which yields the sequence of "negafibonacci" numbers[20] satisfying
Thus the bidirectional sequence is
F−8
F−7
F−6
F−5
F−4
F−3
F−2
F−1
F0
F1
F2
F3
F4
F5
F6
F7
F8
−21
13
−8
5
−3
2
−1
1
0
1
1
2
3
5
8
13
21
Use in mathematics
The Fibonacci numbers are the sums of the "shallow" diagonals (shown in red) of Pascal's triangle.
These numbers also give the solution to certain enumerative problems.[22] The most common is that of counting the number of compositions of 1s and 2s which sum to a given total n: 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:
The Fibonacci numbers can be found in different ways among the set of binarystrings, or equivalently, among the subsets of a given set.
The number of binary strings of length n without consecutive 1s is the Fibonacci number 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. By symmetry, the number of strings of length n without consecutive 0s is also Fn+2. Equivalently, Fn+2 is the number of subsets S ⊂ {1,...,n} without consecutive integers: {i, i+1} ⊄ S for every i. The symmetric statement is: Fn+2 is the number of subsets S ⊂ {1,...,n} without two consecutive skipped integers: that is, S = {a1 < … < ak} with ai+1 ≤ ai + 2.
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 ⊂ {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.
Since , this formula can also be written as
To see this,[25] note that φ and ψ are both solutions of the equations
so the powers of φ and ψ satisfy the Fibonacci recursion. In other words,
and
It follows that for any values a and b, the sequence defined by
satisfies the same recurrence
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:
which has solution
producing the required formula.
Taking the starting values U0 and U1 to be arbitrary constants, a more general solution is:
where
.
Computation by rounding
Since
for all n ≥ 0, the number Fn is the closest integer to . Therefore, it can be found by rounding, that is by the use of the nearest integer function:
Similarly, if we already know that the number F > 1 is a Fibonacci number, we can determine its index within the sequence by
where can be computed using logarithms to other usual bases.
For example, .
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 .[26][27]
This convergence holds regardless of the starting values, excluding 0 and 0, or any pair in the conjugate golden ratio . This can be derived from 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, …, etc. The ratio of
consecutive terms in this sequence shows the same convergence towards
the golden ratio.
Another consequence is that the limit of the ratio of two
Fibonacci numbers offset by a particular finite deviation in index
corresponds to the golden ratio raised by that deviation. Or, in other
words:
Animated
GIF file showing successive tilings of the plane, and a graph of
approximations to the Golden Ratio calculated by dividing successive
pairs of Fibonacci numbers, one by the other. Uses the Fibonacci numbers
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144
Decomposition of powers of the golden ratio
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 n.
This expression is also true for n < 1 if the Fibonacci sequence Fn is extended to negative integers using the Fibonacci rule
Matrix form
A 2-dimensional system of linear difference equations that describes the Fibonacci sequence is
alternatively denoted
which yields . The eigenvalues of the matrix A are and corresponding to the respective eigenvectors
and
As the initial value is
it follows that the nth term is
From this, the nth element in the Fibonacci series
may be read off directly as a closed-form expression:
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:
Taking the determinant of both sides of this equation yields Cassini's identity,
Moreover, since AnAm = 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),
In particular, with m = n,
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).[28]
Recognizing Fibonacci numbers
The question may arise whether a positive integer x is a Fibonacci number. This is true if and only if one or both of or is a perfect square.[29] This is because Binet's formula above can be rearranged to give
,
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
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.[30] In symbols:
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:
and
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.[31]
A different trick may be used to prove
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 note that 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
The sequence is also considered using the symbolic method.[32] More precisely, this sequences corresponds to a specifiable combinatorial class. The specification of this sequence is . Indeed, as stated above, the -th Fibonacci numbes equals the number of way to partition using segments of size 1 or 2.
It follows that the ordinary generating function of the Fibonacci sequence, i.e. , is the complex function .
Other identities
Numerous other identities can be derived using various methods. Some of the most noteworthy are:[33]
This series is convergent for and its sum has a simple closed-form:[34]
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 above closed form.
If x is the reciprocal of an integer k that is greater than 1, the closed form of the series becomes
In particular,
for all positive integers m.
Some math puzzle-books present as curious the particular value that comes from m=1, which is [35] Similarly, m=2 gives
Reciprocal sums
Infinite sums over reciprocal Fibonacci numbers can sometimes be evaluated in terms of theta functions. For example, we can write the sum of every odd-indexed reciprocal Fibonacci number as
and the sum of squared reciprocal Fibonacci numbers as
If we add 1 to each Fibonacci number in the first sum, there is also the closed form
and there is a nested sum of squared Fibonacci numbers giving the reciprocal of the golden ratio,
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 and more generally, every kth 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[38][39]
Any three consecutive Fibonacci numbers are pairwise coprime, which means that, 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 Fp − 1, and if p is congruent to 2 or 3 (mod 5), then, p divides Fp + 1. The remaining case is that p = 5, and in this case p divides Fp.
These cases can be combined into a single formula, using the Legendre symbol:[40]
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 large—say a 500-bit number—then we can calculate Fm (mod n) efficiently using the matrix form. Thus
Fibonacci primes with thousands of digits have been found, but it is not known whether there are infinitely many.[42] Fkn is divisible by Fn, so, apart from F4 = 3, any Fibonacci prime must have a prime index. As there are arbitrarily long runs of composite numbers, there are therefore also arbitrarily long runs of composite Fibonacci numbers.
No Fibonacci number greater than F6 = 8 is one greater or one less than a prime number.[43]
The only nontrivial square Fibonacci number is 144.[44] Attila Pethő proved in 2001 that there is only a finite number of perfect power Fibonacci numbers.[45] In 2006, Y. Bugeaud, M. Mignotte, and S. Siksek proved that 8 and 144 are the only such non-trivial perfect powers.[46]
1, 3, 21, 55 are the only triangular Fibonacci numbers, which was conjectured by Vern Hoggatt and proved by Luo Ming.[47]
Prime divisors of Fibonacci numbers
With the exceptions of 1, 8 and 144 (F1 = F2, F6 and F12) every Fibonacci number has a prime factor that is not a factor of any smaller Fibonacci number (Carmichael's theorem).[48] As a result, 8 and 144 (F6 and F12) are the only Fibonacci numbers that are the product of other Fibonacci numbers OEIS: A235383.
The divisibility of Fibonacci numbers by a prime p is related to the Legendre symbol which is evaluated as follows:
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:[51]
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 Fn are congruent to 1 modulo 4, implying that all odd divisors of Fn (as the products of odd prime divisors) are congruent to 1 modulo 4.[52]
For example,
All known factors of Fibonacci numbers F(i) for all i < 50000 are collected at the relevant repositories.[53][54]
If the members of the Fibonacci sequence are taken mod n, the resulting sequence is periodic with period at most 6n.[55] The lengths of the periods for various n form the so-called Pisano periodsOEIS: A001175.
Determining a general formula for the Pisano periods is an open
problem, which includes as a subproblem a special instance of the
problem of finding the multiplicative order of a modular integer or of an element in a finite field. However, for any particular n, the Pisano period may be found as an instance of cycle detection.
Right triangles
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.
The length of the longer leg of this triangle is equal to the sum of
the three sides of the preceding triangle in this series of triangles,
and the shorter leg is equal to the difference between the preceding
bypassed Fibonacci number and the shorter leg of the preceding triangle.
The first triangle in this series has sides of length 5, 4, and
3. Skipping 8, the next triangle has sides of length 13, 12 (5 + 4 + 3),
and 5 (8 − 3). Skipping 21, the next triangle has sides of length 34,
30 (13 + 12 + 5), and 16 (21 − 5). This series continues indefinitely.
The triangle sides a, b, c can be calculated directly:
These formulas satisfy for all n, but they only represent triangle sides when n > 2.
Any four consecutive Fibonacci numbers Fn, Fn+1, Fn+2 and Fn+3 can also be used to generate a Pythagorean triple in a different way:[56]
Example 1: let the Fibonacci numbers be 1, 2, 3 and 5. Then:
Magnitude
Since Fn is asymptotic to , the number of digits in Fn 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 Fn is asymptotic to .
Applications
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.[57]
Brasch et al. 2012 show how a generalised Fibonacci sequence also can be connected to the field of economics.[58]
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 solvingHilbert's tenth problem.[59]
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.
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 (software development) 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, uses Fibonacci numbers.[60]
The Fibonacci number series is used for optional lossy compression in the IFF8SVX audio file format used on Amiga computers. The number series compands the original audio wave similar to logarithmic methods such as µ-law.[61][62]
Since the conversion
factor 1.609344 for miles to kilometers is close to the golden ratio
(denoted φ), 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.[63]
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,[11] in two consecutive Fibonacci numbers, such as branching in trees, arrangement of leaves on a stem, the fruitlets of a pineapple,[12] the flowering of artichoke, an uncurling fern and the arrangement of a pine cone,[13] and the family tree of honeybees.[64] However, numerous poorly substantiated claims of Fibonacci numbers or golden sections
in nature are found in popular sources, e.g., relating to the breeding
of rabbits in Fibonacci's own unrealistic example, the seeds on a
sunflower, the spirals of shells, and the curve of waves.[65] 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.[66]
Illustration of Vogel's model for n = 1 … 500
A model for the pattern of florets in the head of a sunflower was proposed by H. Vogel in 1979.[67] This has the form
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. It is often said that sunflowers and
similar arrangements have 55 spirals in one direction and 89 in the
other (or some other pair of adjacent Fibonacci numbers), but this is
true only of one range of radii, typically the outermost and thus most
conspicuous.[68]
The bee ancestry code
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.[69] This is under the unrealistic assumption that the ancestors at each level are otherwise unrelated.
The human X chromosome inheritance tree
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".[70])
Luke Hutchison noticed that the number of possible ancestors on the X chromosome inheritance line at a given ancestral generation also follows the Fibonacci sequence.[70] 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 (), 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. (Note that 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 Fibonacci sequence has been generalized in many ways. These include:
Generalizing the index to negative integers to produce the negafibonacci numbers.
Generalizing the index to real numbers using a modification of Binet's formula.[33]
Starting with other integers. Lucas numbers have L1 = 1, L2 = 3, and Ln = Ln−1 + Ln−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 = 2Pn − 1 + Pn − 2.
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.[71]
Adding other objects than integers, for example functions or strings – one essential example is Fibonacci polynomials.
Ball, Keith M (2003), "8: Fibonacci's Rabbits Revisited", Strange Curves, Counting Rabbits, and Other Mathematical Explorations, Princeton, NJ: Princeton University Press, ISBN978-0-691-11321-0.
Beck, Matthias; Geoghegan, Ross (2010), The Art of Proof: Basic Training for Deeper Mathematics, New York: Springer, ISBN978-1-4419-7022-0.
Pisano, Leonardo (2002), Fibonacci's Liber Abaci: A Translation into Modern English of the Book of Calculation, Sources and Studies in the History of Mathematics and Physical Sciences, Sigler, Laurence E, trans, Springer, ISBN978-0-387-95419-6
Singh, Parmanand (1985), "The So-called Fibonacci numbers in ancient and medieval India", Historia Mathematica, 12 (3): 229–44, doi:10.1016/0315-0860(85)90021-7
Knuth, Donald (2006), The Art of Computer Programming, 4. Generating All Trees – History of Combinatorial Generation, Addison–Wesley, p. 50, ISBN978-0-321-33570-8, it
was natural to consider the set of all sequences of [L] and [S] that
have exactly m beats. ...there are exactly Fm+1 of them. For example
the 21 sequences when m = 7 are: [gives list]. In this way
Indian prosodists were led to discover the Fibonacci sequence, as we
have observed in Section 1.2.8 (from v.1)
Wolfram, Stephen (2002). A New Kind of Science. Wolfram Media, Inc. p. 891. ISBN978-1-57955-008-0.
Jones, Judy; Wilson, William (2006), "Science", An Incomplete Education, Ballantine Books, p. 544, ISBN978-0-7394-7582-9
Brousseau, A (1969), "Fibonacci Statistics in Conifers", Fibonacci Quarterly (7): 525–32
Knuth, Donald (1968), The Art of Computer Programming, 1, Addison Wesley, ISBN978-81-7758-754-8, Before
Fibonacci wrote his work, the sequence Fn had already been discussed by
Indian scholars, who had long been interested in rhythmic patterns...
both Gopala (before 1135 AD) and Hemachandra (c. 1150) mentioned the
numbers 1,2,3,5,8,13,21 explicitly [see P. Singh Historia Math 12 (1985)
229–44]" p. 100 (3d ed)...
Agrawala, VS (1969), Pāṇinikālīna Bhāratavarṣa (Hn.). Varanasi-I: TheChowkhamba Vidyabhawan, SadgurushiShya
writes that Pingala was a younger brother of Pāṇini [Agrawala 1969,
lb]. There is an alternative opinion that he was a maternal uncle of
Pāṇini [Vinayasagar 1965, Preface, 121]. … Agrawala [1969, 463–76],
after a careful investigation, in which he considered the views of
earlier scholars, has concluded that Pāṇini lived between 480 and 410 BC
Velankar, HD (1962), 'Vṛttajātisamuccaya' of kavi Virahanka, Jodhpur: Rajasthan Oriental Research Institute, p. 101, "For
four, variations of meters of two [and] three being mixed, five
happens. For five, variations of two earlier – three [and] four, being
mixed, eight is obtained. In this way, for six, [variations] of four
[and] of five being mixed, thirteen happens. And like that, variations
of two earlier meters being mixed, seven morae [is] twenty-one. In this
way, the process should be followed in all mātrā-vṛttas
Gardner, Martin (1996), Mathematical Circus, The Mathematical Association of America, p. 153, ISBN978-0-88385-506-5, It
is ironic that Leonardo, who made valuable contributions to
mathematics, is remembered today mainly because a 19th-century French
number theorist, Édouard Lucas... attached the name Fibonacci to a
number sequence that appears in a trivial problem in Liber abaci
Knott, R, "Fib table", Fibonacci, UK: Surrey has the first 300 Fn factored into primes and links to more extensive tables.
Knuth, Donald (2008-12-11), "Negafibonacci Numbers and the Hyperbolic Plane", Annual meeting, The Fairmont Hotel, San Jose, CA: The Mathematical Association of America
Ribenboim, Paulo (2000), My Numbers, My Friends, Springer-Verlag
Su, Francis E (2000), "Fibonacci GCD's, please", Mudd Math Fun Facts, et al, HMC
Williams, H. C. (1982), "A note on the Fibonacci quotient ", Canadian Mathematical Bulletin, 25 (3): 366–370, doi:10.4153/CMB-1982-053-0, MR0668957. Williams calls this property "well known".
Prime Numbers, Richard Crandall, Carl Pomerance, Springer, second edition, 2005, p.142.
Freyd, Peter; Brown, Kevin S. (1993), "Problems and Solutions: Solutions: E3410", The American Mathematical Monthly, 99 (3): 278–279, doi:10.2307/2325076, JSTOR2325076
Koshy, Thomas (2007), Elementary number theory with applications, Academic Press, p. 581, ISBN978-0-12-372487-8
Brasch, T. von; Byström, J.; Lystad, L.P. (2012), "Optimal Control and the Fibonacci Sequence", Journal of Optimization Theory and Applications, 154 (3): 857–78, doi:10.1007/s10957-012-0061-2, hdl:11250/180781
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