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Fibonacci numbers

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In mathematics, the Fibonacci numbers form a sequence defined recursively by:

 
  F_n := F(n):=
  \begin{cases}
    0             & \mbox{if } n = 0; \\
    1             & \mbox{if } n = 1; \\
    F(n-1)+F(n-2) & \mbox{if } n > 1. \\
   \end{cases}

In other words, one starts with 0 and 1, and then produces the next Fibonacci number by adding the two previous Fibonacci numbers. The first Fibonacci numbers (sequence A000045 in OEIS)

for n = 0, 1, … are
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, …
FibonacciBlocks

A tiling with Fibonacci number-sized squares

OriginsEdit

The earliest known reference to Fibonacci numbers is contained in a book on meters called Chhandah-shāstra (500 BC) by an Indian mathematician named Pingala. As documented by Donald Knuth in The Art of Computer Programming, this sequence was described by the Indian mathematicians Gopala and Hemachandra in 1150, who were investigating the possible ways of exactly bin packing items of length 1 and 2. In the West, it was first studied by Leonardo of Pisa, who was also known as Fibonacci (c. 1200), to describe the growth of an idealised (although biologically unrealistic) rabbit population. The numbers describe the number of pairs in the rabbit population after n months if it is assumed that:

  • in the first month there is just one newly-born pair,
  • new-born pairs become fertile from their second month on
  • each month every fertile pair begets a new pair, and
  • the rabbits never die

Suppose that in month n we have a total of a pairs of rabbits and in month n + 1 we have b pairs. In month n + 2 we will necessarily have a + b pairs, because all a pairs of rabbits from month n will be fertile and produce a new pairs of offspring (since all a rabbits are at least two months old) -- plus b, which are the existing pairs of rabbits at n + 1 (remember the assumption that no rabbit ever dies).

The bee ancestry codeEdit

Fibonacci is also stated as having described the sequence "encoded in the ancestry of a male bee." This turns out to be the Fibonacci sequence. One can derive this truth by taking the following facts:

  • If an egg is laid by a single female, it hatches a male.
  • If, however, the egg is fertilized by a male, it hatches a female.
  • Thus, a male bee will always have one parent, and a female bee will have two.

If one traces the ancestry of this male bee (1 bee), he has 1 female parent (1 bee). This female had 2 parents, a male and a female (2 bees). The female had two parents, a male and a female, and the male had one female (3 bees). Those two females each had two parents, and the male had one (5 bees). If one continues this sequence, it gives a perfectly accurate depiction of the Fibonacci sequence.

However, this statement is mostly theoretical. In reality, some ancestors of a particular bee will always be sisters or brothers, thus breaking the lineage of distinct parents.

Relation to the golden ratioEdit

Limit of consecutive quotientsEdit

As was pointed out by Johannes Kepler, the ratio of consecutive Fibonacci numbers, that is:

\frac{F(n+1)}{F(n)},

converges to the golden ratio \varphi (phi), defined as the positive solution of the equation:

\frac{x}{1}=\frac{1}{x-1} or equivalently x=1+\frac{1}{x}

Proof (assuming that the ratio of successive terms converges to a limit):

x={\lim_{n\to\infty}\frac{F(n+1)}{F(n)}}

=\lim_{n\to\infty}\frac{F(n)+F(n-1)}{F(n)}

=\lim_{n\to\infty}\left(\frac{F(n)}{F(n)}+\frac{F(n-1)}{F(n)}\right)

=1+\lim_{n\to\infty}\frac{F(n-1)}{F(n)}

=1+\frac{1}{\displaystyle\lim_{n\to\infty}\frac{F(n)}{F(n-1)}}

=1+\frac1x

Closed form expressionEdit

Like every sequence defined by linear recursion, the Fibonacci numbers have a closed-form solution. It has become known as Binet's formula:

F\left(n\right) = {{\varphi^n-(1-\varphi)^n} \over {\sqrt 5}}

where \varphi=(1+\sqrt 5)/2\,\! is the golden ratio, defined as the positive root of the equation:

x^2-x-1=0\,\!, also known as the generating polynomial of the similar Fibonacci recursion:
F(n+2)-F(n+1)-F(n)=0\,\!

Proof (by induction):

Note that by definition \varphi+1=\varphi^2\,\! and that the other root of the equation is 1-\varphi\,\!, so that also (1-\varphi)+1=(1-\varphi)^2\,\!. Then

F(0)=(1-1)/\sqrt 5=0\,\! and
F(1)=(\varphi-(1-\varphi))/\sqrt 5=(-1+2\varphi)/\sqrt 5=(-1+(1+\sqrt 5))/\sqrt 5=1\,\!,

establishing the base cases of the induction, which itself follows from:

F(n)+F(n-1)=\,\!
=\frac{\varphi^n-(1-\varphi)^n}{\sqrt 5}+\frac{\varphi^{n-1}-(1-\varphi)^{n-1}}{\sqrt 5}
=\frac{(\varphi+1)\varphi^{n-1}-((1-\varphi)+1)(1-\varphi)^{n-1}}{\sqrt 5}
=\frac{\varphi^2\varphi^{n-1}-(1-\varphi)^2(1-\varphi)^{n-1}}{\sqrt 5}
=\frac{\varphi^{n+1}-(1-\varphi)^{n+1}}{\sqrt 5}
=F(n+1)\,\!

Matrix formEdit

A 2-dimensional system of linear difference equations that describes the Fibonacci sequence is

{F_{k+2} \choose F_{k+1}} = \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix} {F_{k+1} \choose F_{k}}

or

\vec F_{k+1} = A \vec F_{k}

The eigenvalues of the matrix A are \varphi\,\! and (1-\varphi)\,\!, and the elements of the eigenvectors of A, {\varphi \choose 1} and {1 \choose -\varphi}, are in the ratios \varphi\,\! and (1-\varphi\,\!).

Note that this matrix 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 mean: \varphi\,\! = [1; 1, 1, 1, 1, …]. The Fibonacci numbers occur as the ratio of successive convergents of the continued fraction for \varphi\,\!, and the matrix formed from successive convergents of any continued fraction has a determinant of +1 or −1.

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The matrix representation gives the following closed expression for the Fibonacci numbers:

\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 the identity:

 F_{n+1}F_{n-1} - F_n^2 = (-1)^n

Additionally, since  A^n A^m=A^{m+n} for any square matrix A, the following identities can be derived:

{F_n}^2 + {F_{n-1}}^2 = F_{2n-1}
F_{n+1}F_{m} + F_n F_{m-1} = F_{m+n}\,

ComputationEdit

Several algorithms exist for computing Fibonacci numbers, and these are commonly used as examples in introductory computer science courses for explaining recursion or as examples of exponential vs. polynomial time complexity.

The most straightforward algorithm mirrors the recursive definition of the sequence. In pseudocode:

function fib(n)
    if n = 0 return 0
    if n = 1 return 1
    return fib(n-1) + fib(n-2)

This algorithm runs in O(Fn) = O(φn), i.e. exponential time (and O(n) linear space) -- thus, it is extremely inefficient for calculating large Fibonacci numbers. It can be brought down to linear time by using memoization in the algorithm.

A more efficient iterative algorithm computes the Fibonacci numbers "from the bottom up", starting with the two values 0 and 1, and then repeatedly replacing the first number by the second, and the second number by the sum of the two. This runs in O(n) time and O(1) (constant) space:

function fib(n)
    var a,b := 0,1
    repeat n times
        a,b := b,a+b
    return a

A logarithmic time algorithm can be derived from the matrix form:

function fib(n)
    if n = 0
        return 0
    else
        var a := expt([[1,1],[1,0]], n)
        return a[2]

(* matrix exponentiation in O(log n) *)
function expt(a,n)
    if n = 1
        return a
    else if even(n)
        return expt(mult(a,a), n/2)
    else
        return mult(a, expt(a,n-1))

(* multiplication of 2×2 matrices *)
function mult([[a,b],[c,d]], [[w,x],[y,z]])
    return [[a*w+b*y, a*x+b*z], [c*w+d*y, c*x+d*z]]

(Note that, in every case, complexity is given in terms of the value n of the input. If we measure it in terms of the size of the input, which is m = \log_2 n bits in the case of an integer n, then the complexity of the naïve recursive algorithm becomes O(\varphi^{2^m}), of the second algorithm O(2^m), and of the final algorithm O(m).)

Since a closed formula exists, the practical use of these algorithms is limited to cases where exact values of F_n are needed for large n. When computing using the closed form, rounding errors will inevitably occur and grow as n becomes larger, because \varphi, being irrational, must be represented by a floating point number. Note that with standard IEEE 64 bit floats, results will not be precise already at n=75. However, similarly, using standard 32 bit integers will run into an overflow situation before that.

ApplicationsEdit

The Fibonacci numbers are important in the 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.

Yuri Matiyasevich was able to show that the Fibonacci numbers can be defined by a Diophantine equation, which led to his original solution of Hilbert's tenth problem.

The Fibonacci numbers occur in a formula about the diagonals of Pascal's triangle (see binomial coefficient).

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.

Fibonacci numbers are also used by some pseudorandom number generators.

In music Fibonacci numbers are sometimes used to determine tunings, and, as in visual art, to determine the length or size of content or formal elements. Examples include Béla Bartók's Music for Strings, Percussion, and Celesta. In addition, the syllables of the lyrics of parts of the Tool song Lateralus follow the Fibonacci sequence in each line, for instance "Black/Then/White are/All I see/In my infancy/Red and yellow then came to be".

Since the conversion factor 1.609 for miles to kilometers is close to the golden mean 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 base φ being shifted. To go from kilometers to miles shift the register down the Fibonacci sequence instead.

Fibonacci numbers in natureEdit

Brocolli

Romanesco Broccoli clearly illustrating the fractal repeating spiral pattern found within nature

Fibonacci sequences have been noted to appear in biological settings, such as the branching patterns of leaves in grasses and flowers, branching in bushes and trees, the arrangement of pines on a pine cone, seeds on a raspberry, and spiral patterns in horns and shells. The scales on the surface of a pineapple are arranged in two interlocking spirals, eight spirals in one direction, thirteen in the other; each being a Fibonacci number. Przemyslaw Prusinkiewicz has advanced the idea that these can be in part understood as the expression of certain algebraic constraints on free groups, specifically as certain Lindenmayer grammars. Generally one sees Fibonacci numbers arise in the study of the fractal Fuchsian groups and Kleinian groups, and systems that possess such symmetries. For example, the solutions to reaction-diffusion differential equations (such as that seen in the Belousov-Zhabotinsky reaction) can show such a patterning; in biology, genes often express themselves through gene regulatory networks, that is, in terms of several enzymes controlling a reaction, which can be modelled with reaction-diffusion equations. Such systems rarely give the Fibonacci sequence exactly or directly; rather, the relationship occurs deeper in the theory. Similar patterns also occur in non-biological systems, such as in sphere packing models.

IdentitiesEdit

F(n + 1) = F(n) + F(n − 1)
F(0) + F(1) + F(2) + … + F(n) = F(n + 2) − 1
F(1) + 2 F(2) + 3 F(3) + … + n F(n) = n F(n + 2) − F(n + 3) + 2

These identities can be proven using many different methods. But, among all, we wish to present an elegant proof for each of them using combinatorial arguments here. In particular, F(n) can be interpreted as the number of ways summing 1's and 2's to n − 1, with the convention that F(0) = 0, meaning no sum will add up to −1, and that F(1) = 1, meaning the empty sum will "add up" to 0. Here the order of the summands matters. For example, 1 + 2 and 2 + 1 are considered two different sums and are counted twice.

Proof of the first identity. Without loss of generality, we may assume n ≥ 1. Then F(n + 1) counts the number of ways summing 1's and 2's to n.

When the first summand is 1, there are F(n) ways to complete the counting for n − 1; and the first summand is 2, there are F(n − 1) ways to complete the counting for n − 2. Thus, in total, there are F(n) + F(n − 1) ways to complete the counting for n. \Box

Proof of the second identity. We count the number of ways summing 1's and 2's to n + 1 such that at least one of the summands is 2.

As before, there are F(n + 2) ways summing 1's and 2's to n + 1 when n ≥ 0. Since there is only one sum of n + 1 that does not use any 2, namely 1 + … + 1 (n + 1 terms), we subtract 1 from F(n + 2).

Equivalently, we can consider the first occurrence of 2 as a summand. If, in a sum, the first summand is 2, then there are F(n) ways to the complete the counting for n − 1. If the second summand is 2 but the first is 1, then there are F(n − 1) ways to complete the counting for n − 2. Proceed in this fashion. Eventually we consider the (n + 1)th summand. If it is 2 but all of the previous n summands are 1's, then there are F(0) ways to complete the counting for 0. If a sum contains 2 as a summand, the first occurrence of such summand must take place in between the first and (n + 1)th position. Thus F(n) + F(n − 1) + … + F(0) gives the desired counting. \Box

Proof of the third identity. This identity can be established in two stages. First, we count the number of ways summing 1s and 2s to −1, 0, …, or n + 1 such that at least one of the summands is 2.

By our second identity, there are F(n + 2) − 1 ways summing to n + 1; F(n + 1) − 1 ways summing to n; …; and, eventually, F(2) − 1 way summing to 1. As F(1) − 1 = F(0) = 0, we can add up all n + 1 sums and apply the second identity again to obtain

   [F(n + 2) − 1] + [F(n + 1) − 1] + … + [F(2) − 1]
= [F(n + 2) − 1] + [F(n + 1) − 1] + … + [F(2) − 1] + [F(1) − 1] + F(0)
= F(n + 2) + [F(n + 1) + … + F(1) + F(0)] − (n + 2)
= F(n + 2) + F(n + 3) − (n + 2).

On the other hand, we observe from the second identity that there are

  • F(0) + F(1) + … + F(n − 1) + F(n) ways summing to n + 1;
  • F(0) + F(1) + … + F(n − 1) ways summing to n;

……

  • F(0) way summing to −1.

Adding up all n + 1 sums, we see that there are

  • (n + 1) F(0) + n F(1) + … + F(n) ways summing to −1, 0, …, or n + 1.

Since the two methods of counting refer to the same number, we have

(n + 1) F(0) + n F(1) + … + F(n) = F(n + 2) + F(n + 3) − (n + 2)

Finally, we complete the proof by subtracting the above identity from n + 1 times the second identity. \Box

Common factors Edit

Any two consecutive Fibonacci numbers are relatively prime. Suppose that Fn and Fn+1 have a common factor g. Then Fn−1 = Fn+1Fn must also be a multiple of g; and by induction the same must be true of all lower Fibonacci numbers. But F1 = 1, so g = 1.

Other identities include relationships to the Lucas numbers, which have the same recursive properties but start with L0=2 and L1=1. These properties include F2n=FnLn

Power seriesEdit

The Fibonacci power series

s(x)=\sum_{n=1}^\infty F(n) x^n

has a simple and interesting closed-form solution for x < 1/φ:

s(x)=\frac{x}{1-x-x^2}.

This function is therefore the generating function of the Fibonacci sequence. It can be proven as follows:

s(x) = F_0 + F_1x + F_2x^2 + \cdots = \sum_{k=0}^{\infty} F_k x^k

Substituting F_k = F_{k-1} + F_{k-2}:

s(x) = F_0 + F_1x + \sum_{k=2}^{\infty} \left( F_{k-1} + F_{k-2} \right) x^k
= x + \sum_{k=2}^\infty F_{k-1} x^k + \sum_{k=2}^\infty F_{k-2} x^k
= x + x \sum_{k=2}^\infty F_{k-1} x^{k-1} + x^2 \sum_{k=2}^\infty F_{k-2} x^{k-2}
= x + x \sum_{j=1}^\infty F_j x^j + x^2 \sum_{m=0}^\infty F_m x^m
= x + x \left( \sum_{j=0}^\infty F_j x^j - F_0 \right) + x^2 s(x)
= x + x s(x) + x^2 s(x)

Therefore,

s(x) = \frac{x}{1 - x - x^2}

In particular, math puzzle-books note the curious value \frac{s(\frac{1}{10})}{10}=\frac{1}{89}. The sum is easily proved by noting that

s+\frac{s}{x} = 1 + \sum_{n=1}^\infty (F(n)+F(n+1)) x^n

and then explictly evaluating the sum.

Reciprocal sum constantEdit

The sum of the reciprocals of the positive Fibonacci numbers converges: 
C = \sum_{n=1}^{\infty} F(n)^{-1} = 3.359885 \dots
A079586


Convergence is easy to show with the ratio test.

This value has been proven irrational by André-Jeannin, R. No closed form is currently known.

See also the Mathworld article on the subject.

GeneralizationsEdit

Vector spaceEdit

The term Fibonacci sequence is also applied more generally to any function g where g(n + 2) = g(n) + g(n + 1). These functions are precisely those of the form g(n) = aF(n) + bF(n + 1) for some numbers a and b, so the Fibonacci sequences form a vector space with the functions F(n) and F(n + 1) as a basis.

Similar integer sequencesEdit

Lucas numbers Edit

In particular, the Fibonacci sequence L with L(1) = 1 and L(2) = 3 is referred to as the Lucas numbers, after Edouard Lucas. This sequence was described by Leonhard Euler in 1748, in the Introductio in Analysin Infinitorum. The significance in the Lucas numbers L(n) lies in the fact that raising the golden ratio to the nth power yields:

\left( \frac 1 2 \left( 1 + \sqrt{5} \right) \right)^n = \frac 1 2 \left( L(n) + F(n) \sqrt{5} \right)

Lucas numbers are related to Fibonacci numbers by the relation:

L\left(n\right)=F\left(n-1\right)+F\left(n+1\right)

A generalization of the Fibonacci sequence are the Lucas sequences. One kind can be defined thus:

U(0) = 0
U(1) = 1
U(n + 2) = PU(n + 1) − QU(n)

where the normal Fibonacci sequence is the special case of P = 1 and Q = −1. Another kind of Lucas sequence begins with V(0) = 2, V(1) = P. Such sequences have applications in number theory and primality proving.

The Padovan sequence is generated by the recurrence P(n) = P(n − 2) + P(n − 3).

Tribonacci numbers Edit

The tribonacci numbers are like the Fibonacci numbers, but instead of starting with two predetermined terms, the sequence starts with three predetermined terms and each term afterwards is the sum of the preceding three terms. The first few tribonacci numbers are A000073

1, 1, 2, 4, 7, 13, 24, 44, 81, 149, 274, 504, 927, 1705, 3136, 5768, 10609, 19513, 35890, 66012, …

The tribonacci constant is the ratio toward which adjacent tribonacci numbers tend. It is a root of the polynomial x3 − x2 − x − 1, approximately 1.83929, and also satisfies the equation x + x−3 = 2. It is important in the study of the snub cube.

Tetranacci numbers Edit

The tetranacci numbers start with four predetermined terms, each term afterwards being the sum of the preceding four terms. The first few tetranacci numbers are A000078

1, 1, 2, 4, 8, 15, 29, 56, 108, 208, 401, 773, 1490, 2872, 5536, 10671, 20569, 39648, 76424, 147312, 283953, 547337, …

The tetranacci constant is the ratio toward which adjacent tetranacci numbers tend. It is a root of the polynomial x4x3x2x − 1, approximately 1.92756, and also satisfies the equation x + x−4 = 2.

Other ...anacci numbers Edit

Pentanacci, hexanacci and heptanacci numbers have been computed, but they have not interested researchers much.

Other generalizationsEdit

The Fibonacci polynomials are another generalization of Fibonacci numbers.

A random Fibonacci sequence can be defined by tossing a coin for each position n of the sequence and taking F(n)=F(n-1)+F(n-2) if it lands heads and F(n)=F(n-1)-F(n-2) if it lands tails. Work by Furstenburg and Kesten guarantees that this sequence almost surely grows exponentially at a constant rate: the constant is independent of the coin tosses and was computed in 1999 by Divakar Viswanath. It is now known as Viswanath's constant.

A repfigit or Keith number is an integer, that when its digits start a Fibonacci sequence with that number of digits, the original number is eventually reached. An example is 47, because the Fibonacci sequence starting with 4 and 7 (4,7,11,18,29,47) reaches 47. A repfigit can be a tribonacci sequence if there are 3 digits in the number, a tetranacci number if the number has four digits, etc. The first few repfigits are A007629

14, 19, 28, 47, 61, 75, 197, 742, 1104, 1537, 2208, 2580, 3684, 4788, 7385, 7647, 7909, …

Since the set of sequences satisfying the relation S(n) = S(n-1) + S(n-2) is closed under termwise addition and under termwise multiplication by a constant, it can be viewed as a vector space. Any such sequence is uniquely determined by a choice of two elements, so the vector space is two-dimensional. If we abbreviate such a sequence as (S(0), S(1)), the Fibonacci sequence F(n) = (0, 1) and the shifted Fibonacci sequence F(n-1) = (1, 0) are seen to form a canonical basis for this space, yielding the identity:

S(n) = S(0)F(n-1) + S(1)F(n)

for all such sequences S. For example, if S is the Lucas sequence 1, 3, 4, 7, 11…, then we obtain L(n) = F(n-1) + 3F(n).

Fibonacci primesEdit

The first few Fibonacci numbers that are also prime numbers are A005478

2, 3, 5, 13, 89, 233, 1597, 28657, 514229, …. It seems likely that there are infinitely many Fibonacci primes, but this has yet to be proven.

Fibonacci strings Edit

In analogy to its numerical counterpart, a Fibonacci string is defined by:

 
  F_n := F(n):=
  \begin{cases}
    b             & \mbox{if } n = 0; \\
    a             & \mbox{if } n = 1; \\
    F(n-1)+F(n-2) & \mbox{if } n > 1. \\
   \end{cases}
 ,

where + denotes the concatenation of two strings. The sequence of Fibonacci strings starts:

b, a, ab, aba, abaab, abaababa, abaababaabaab, …

The length of each Fibonacci string is a Fibonacci number, and similarly there exists a corresponding Fibonacci string for each Fibonacci number.

Fibonacci strings appear as inputs for the worst case in some computer algorithms.

Popular cultureEdit

  • The Fibonacci sequence plays a prominent part in the bestselling novel The Da Vinci Code
  • Tool's song "Lateralus" from the album of the same name features the Fibonacci sequence syllabically in the verses of the song. The syllables count from one to eight via the sequence and from eight to one via the sequence.
  • Along with the concepts of the golden rectangle and golden spiral, the Fibonacci sequence is used in Darren Aronofsky's indie film π (1998)
  • Referenced in the film of The Phantom Tollbooth.
  • Used in Steven Spielberg's miniseries Taken.
  • It was also used as a key plot point in an episode of the Disney original television series So Weird.
  • In a FoxTrot comic, Jason and Marcus are playing football. Jason yells, "Hut 0! Hut 1! Hut 1! Hut 2!" all the way until "Hut 13!" in the Fibonacci sequence. Marcus yells, "Is it the Fibonacci sequence?" Jason says, "Correct! Touchdown, Marcus!"
  • BT (Brian Transeau) released a dance track in 2000, entitled the "Fibonacci Sequence," which features a sample of a reading of the sequence.
  • Dr. Steel released a song titled "Fibonacci Sequence" in 2005.
  • The Fibonacci Sequence is a key plot point in the television show MathNet's episode 'The Case of the Willing Parrot'.
  • It is also referenced to in Numb3rs, the television series. Many times the cast reference the fibonacci sequence and the relationship it has with nature to further emphasise the wonders of math.

See alsoEdit

JournalsEdit

ReferencesEdit

External linksEdit

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