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A Fibonacci sequence of order n is an integer sequence in which each sequence element is the sum of the previous elements (with the exception of the first elements in the sequence). The usual Fibonacci numbers are a Fibonacci sequence of order 2.
The following steps describe how to encode a nonzero integer .Note that denotes the Negafibonacci sequence.. If is positive, compute the greatest odd negative integer such that the sum of the odd negative terms of the Negafibonacci sequence from -1 to with a step of -2, is greater than or equal to :
{{For|the chamber ensemble|Fibonacci Sequence (ensemblhello mathematics, the Fibonacci sequence is a sequence in which each element is the sum of the two elements that precede it. Numbers that are part of the Fibonacci sequence are known as Fibonacci numbers , commonly denoted F n .
More precisely, if N is any positive integer, there exist positive integers c i ≥ 2, with c i + 1 > c i + 1, such that = =, where F n is the n th Fibonacci number. Such a sum is called the Zeckendorf representation of N. The Fibonacci coding of N can be derived from its Zeckendorf representation.
To encode an integer N: . Find the largest Fibonacci number equal to or less than N; subtract this number from N, keeping track of the remainder.; If the number subtracted was the i th Fibonacci number F(i), put a 1 in place i − 2 in the code word (counting the left most digit as place 0).
For any integer n, the sequence of Fibonacci numbers F i taken modulo n is periodic. The Pisano period, denoted π ( n ), is the length of the period of this sequence. For example, the sequence of Fibonacci numbers modulo 3 begins:
The coefficients of the Fibonacci polynomials can be read off from a left-justified Pascal's triangle following the diagonals (shown in red). The sums of the coefficients are the Fibonacci numbers. If F ( n , k ) is the coefficient of x k in F n ( x ), namely
Fibonacci instead would write the same fraction to the left, i.e., . Fibonacci used a composite fraction notation in which a sequence of numerators and denominators shared the same fraction bar; each such term represented an additional fraction of the given numerator divided by the product of all the denominators below and to the right of it.