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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.
Numbers that are part of the Fibonacci sequence are known as Fibonacci numbers, commonly denoted F n . Many writers begin the sequence with 0 and 1, although some authors start it from 1 and 1 [ 1 ] [ 2 ] and some (as did Fibonacci) from 1 and 2.
For example, consider the recursive formulation for generating the Fibonacci sequence: F i = F i−1 + F i−2, with base case F 1 = F 2 = 1. Then F 43 = F 42 + F 41, and F 42 = F 41 + F 40. Now F 41 is being solved in the recursive sub-trees of both F 43 as well as F 42. Even though the total number of sub-problems is actually small (only 43 ...
Plot of the first 10,000 Pisano periods. In number theory, the nth Pisano period, written as π (n), is the period with which the sequence of Fibonacci numbers taken modulo n repeats.
Truncating this sequence to k terms and forming the corresponding Egyptian fraction, e.g. (for k = 4) + + + = results in the closest possible underestimate of 1 by any k-term Egyptian fraction. [5] That is, for example, any Egyptian fraction for a number in the open interval ( 1805 / 1806 , 1) requires at least five terms.
Every sequence of positive integers satisfying the Fibonacci recurrence occurs, shifted by at most finitely many positions, in the Wythoff array. In particular, the Fibonacci sequence itself is the first row, and the sequence of Lucas numbers appears in shifted form in the second row ( Morrison 1980 ).
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 example, the problem of computing the Fibonacci sequence exhibits overlapping subproblems. The problem of computing the n th Fibonacci number F ( n ), can be broken down into the subproblems of computing F ( n − 1) and F ( n − 2), and then adding the two.