# Questions tagged [fibonacci-numbers]

Questions on the Fibonacci numbers, a special sequence of integers that satisfy the recurrence $F_n=F_{n-1}+F_{n-2}$ with the initial conditions $F_0=0$ and $F_1=1$.

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### Has anyone invented a computationally simple method to calculate the probability of at least n1-consecutive die rolls, for n2-sided die, n3-rolls?

I think I have invented a formula that allows a computer to very easily calculate the probability of at least n1-consecutive die rolls, on an n2-sided die, rolling it n3-times. For example, for a 3-...
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### Fibonacci sequence into explicit using backtracking method [closed]

How can the fibonacci sequence be converted from recursive to an explicit formula using backtracking?
25 views

### Patterns made of Fibonacci's sequence

I was trying to test music with math, so why not Fibonacci? I'd use it lowering every number of the sequence to 7 or minor, I mean, if a certain number n is greater than 7, I'd do n-1, until getting ...
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### A conjecture concerning Fibonacci

First of all, I'm aware that this conjecture has probably already been made and proved. I was just playing around, and I'd like to know whether it is in fact true and perhaps a hint as to how to prove ...
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### How to prove the following strange relation concerning fibonacci numbers

Is the following relation concerning fibonacci numbers, $F_n$ true? $$F_{2n-1}^n=2^{2n^2}\prod\limits_{r=1}^{n}\prod\limits_{s=1}^{n}\left(\cos^2\frac{r\pi}{2n+1}+\cos^2\frac{s\pi}{2n+1}\right)$$ I ...
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### (why) is this ratio the golden ratio?

Looking at a slight variation of the fibonacci sequence f(x) = f(x-1) + f(x-2) + 1 where f(1) = 1, f(2) = 1 I'm trying to find the ratio of this sequence but can't figure out how. To get an ...