# Questions tagged [fibonacci-numbers]

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### Splitting natural numbers into subsets with sums equal to A066258

Let $F(n)$ be A000045 i.e. Fibonacci numbers. Here $$F(n) = F(n-1) + F(n-2), \\ F(0) = 0, F(1) = 1$$ Let $a(n)$ be A066258 i.e. $$a(n) = F(n)^2F(n+1)$$ Let $b(n)$ be A345253 i.e. maximal ...
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### Slightly modified program for the A345253 such that specific partial sums equal A066258

Let $F(n)$ be A000045 i.e. Fibonacci numbers. Here $$F(n) = F(n-1) + F(n-2), \\ F(0) = 0, F(1) = 1$$ Let $a(n)$ be A345253 i.e. maximal Fibonacci tree: arrangement of the positive integers as ...
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### On the finite sum of reciprocal Fibonacci sequences

I want to check if $$\left\lfloor \left( \sum_{k=n}^{2n}{\frac{1}{F_{2k}}} \right)^{-1} \right\rfloor =F_{2n-1}~~(n\ge 3) \tag{*}$$ where $\lfloor x \rfloor$ is th floor function. The Fibonacci ...
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### Golden ratio base

Let $\phi$ be the golden ratio and look at real numbers as expansions in digits from base $\phi + 1$. Has this base been considered or studied anywhere? Note that integers in this base are palindromes ...
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### Avoiding the Fibonacci numbers

For given positive integers $a$ and $b$, let $(a,b)$ be "special" if $an+b$ is not a Fibonacci number for every positive integer $n$. For instance, $(8,4)$ and $(8,6)$ are special. There are ...
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### Conjecture about primes and Fibonacci numbers

I posted this conjecture on math.stackexchange, but I received no answer proving or disproving it: if $m > 4$ is a positive integer not divisible by $2$ or $3$, it's ever possible to find a ...
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### The Fibonacci sequence modulo $5^n$

Let $(F_k)_{k=0}^\infty$ be the classical Fibonacci sequence, defined by the recursive formula $F_{k+1}=F_k+F_{k-1}$ where $F_0=0$ and $F_1=1$. For every $n\in\mathbb N$ let $\pi(n)$ be the smallest ...
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### Reciprocals of Fibonacci numbers

Is the sum of the reciprocals of Fibonacci numbers a transcendental?