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Fibonacci numbers arise in the analysis of the Fibonacci heap data structure. A one-dimensional optimization method, called the Fibonacci search technique, uses Fibonacci numbers. [74] The Fibonacci number series is used for optional lossy compression in the IFF 8SVX audio file format used on Amiga computers.
The amortized performance of a Fibonacci heap depends on the degree (number of children) of any tree root being (), where is the size of the heap. Here we show that the size of the (sub)tree rooted at any node x {\displaystyle x} of degree d {\displaystyle d} in the heap must have size at least F d + 2 {\displaystyle F_{d+2}} , where F i ...
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.
For generalized Fibonacci sequences (satisfying the same recurrence relation, but with other initial values, e.g. the Lucas numbers) the number of occurrences of 0 per cycle is 0, 1, 2, or 4. The ratio of the Pisano period of n and the number of zeros modulo n in the cycle gives the rank of apparition or Fibonacci entry point of n .
Even though the total number of sub-problems is actually small (only 43 of them), we end up solving the same problems over and over if we adopt a naive recursive solution such as this. Dynamic programming takes account of this fact and solves each sub-problem only once. Figure 2. The subproblem graph for the Fibonacci sequence.
The Leonardo numbers are related to the Fibonacci numbers by the relation () = (+),.. From this relation it is straightforward to derive a closed-form expression for the Leonardo numbers, analogous to Binet's formula for the Fibonacci numbers:
The Fermat test and the Fibonacci test are simple examples, and they are effective when combined. John Selfridge has conjectured that if p is an odd number, and p ≡ ±2 (mod 5), then p will be prime if both of the following hold: 2 p−1 ≡ 1 (mod p), f p+1 ≡ 0 (mod p), where f k is the k-th Fibonacci number. The first condition is the ...
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).