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The Fibonomial coefficients (sequence A010048 in the OEIS) are similar to binomial coefficients and can be displayed in a triangle similar to Pascal's triangle. The first eight rows are shown below. The first eight rows are shown below.
This sequence of numbers of parents is the Fibonacci sequence. The number of ancestors at each level, F n, is the number of female ancestors, which is F n−1, plus the number of male ancestors, which is F n−2. [91] [92] This is under the unrealistic assumption that the ancestors at each level are otherwise unrelated.
A repfigit, or Keith number, is an integer such 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 Lagged Fibonacci generator (LFG or sometimes LFib) is an example of a pseudorandom number generator. This class of random number generator is aimed at being an improvement on the 'standard' linear congruential generator. These are based on a generalisation of the Fibonacci sequence. The Fibonacci sequence may be described by the recurrence ...
Thus it suffices to compute Pisano periods for prime powers =. (Usually, () = (), unless p is k-Wall–Sun–Sun prime, or k-Fibonacci–Wieferich prime, that is, p 2 divides F k (p − 1) or F k (p + 1), where F k is the k-Fibonacci sequence, for example, 241 is a 3-Wall–Sun–Sun prime, since 241 2 divides F 3 (242).)
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).
As in Pascal's triangle and other similarly constructed triangles, [2] sums of components along diagonal paths in Bernoulli's triangle result in the Fibonacci numbers. [3] As the third column of Bernoulli's triangle (k = 2) is a triangular number plus one, it forms the lazy caterer's sequence for n cuts, where n ≥ 2. [4]
In the Fibonacci sequence, each number is the sum of the previous two numbers. Fibonacci omitted the "0" and first "1" included today and began the sequence with 1, 2, 3, ... . He carried the calculation up to the thirteenth place, the value 233, though another manuscript carries it to the next place, the value 377.