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The semi-Fibonacci sequence (sequence A030067 in the OEIS) is defined via the same recursion for odd-indexed terms (+) = + and () =, but for even indices () = (), . The bisection A030068 of odd-indexed terms s ( n ) = a ( 2 n − 1 ) {\displaystyle s(n)=a(2n-1)} therefore verifies s ( n + 1 ) = s ( n ) + a ( n ) {\displaystyle s(n+1)=s(n)+a(n ...
Fibonacci numbers are also strongly related to the golden ratio: Binet's formula expresses the n-th Fibonacci number in terms of n and the golden ratio, and implies that the ratio of two consecutive Fibonacci numbers tends to the golden ratio as n increases.
Here the fibonorial constant (also called the fibonacci factorial constant [1]) is defined by = = (), where = and is the golden ratio. An approximate truncated value of C {\displaystyle C} is 1.226742010720 (see (sequence A062073 in the OEIS ) for more digits).
The Lucas sequence has the same recursive relationship as the Fibonacci sequence, where each term is the sum of the two previous terms, but with different starting values. [1] This produces a sequence where the ratios of successive terms approach the golden ratio , and in fact the terms themselves are roundings of integer powers of the golden ...
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A Fibonacci prime is a Fibonacci number that is prime, a type of integer sequence prime. The first Fibonacci primes are (sequence A005478 in the OEIS ): 2 , 3 , 5 , 13 , 89 , 233 , 1597, 28657, 514229, 433494437, 2971215073, ....
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. Pisano periods are named after Leonardo Pisano, better known as Fibonacci. The existence of periodic functions in Fibonacci numbers was noted by Joseph Louis Lagrange in 1774. [1] [2]
The Fibonacci cube of order n is the simplex graph of the complement graph of an n-vertex path graph. [2] That is, each vertex in the Fibonacci cube represents a clique in the path complement graph, or equivalently an independent set in the path itself; two Fibonacci cube vertices are adjacent if the cliques or independent sets that they represent differ by the addition or removal of a single ...