# Questions tagged [fibonacci-numbers]

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26
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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 ...

2
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0
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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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### Equivalence of primes based on the partition of their Pisano periods

The period of Fibonacci numbers modulo $m$ is called Pisano period and its length is denoted as $\pi(m)$. Define the Pisano partition of $m$ as the set partition of the indices $\{0,1,\dotsc,\pi(m)-1\}...

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### Reference for formula expressing products of two Fibonacci numbers in Zeckendorf-basis

It is well-known folklore that every natural integer has a unique Zeckendorf expansion as a
sum over a finite set of Fibonacci numbers containing no pair of consecutive Fibonacci numbers.
It is easy ...

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1
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### Explicit formula for Fibonacci numbers; compositions of $n$

A Fibonacci-type sequence is a sequence with two seed-values, $F_1$ and $F_2$, and which, for all $n>2$, abides by the recurrence relation $F_n = F_{n-1} + F_{n-2}$. If $F_1 = F_2 = s$, then the $n$...

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2
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### Number of coefficients equal to $k$ in certain "Fibonacci polynomials"

Let $F_i$ denote the $i$th Fibonacci number (with $F_1=F_2=1$). Define
$$ P_n(x) = \prod_{i=1}^n (1+x^{F_{i+1}}). $$
Let $\nu_k(n)$ denote the number of coefficients of the polynomial $P_n(x)$
that ...

3
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1
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### Why doesn't the number of ones in the binary representation of Fibonacci numbers grow linearly? [closed]

I am a third-year computer science student. I am interested, why doesn't the number of ones in the binary representation of Fibonacci numbers grow linearly? I would expect it to grow linearly all the ...

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### Recurrences (based on Fibonacci numbers) for the first differences of numbers filtred by equality of binary functions

First we need to set some binary functions:
Let
$$\ell(n)=\left\lfloor\log_2 n\right\rfloor$$
Let $\operatorname{wt}(n)$ be A000120, i.e., $1$'s-counting sequence: number of $1$'s in binary expansion ...

4
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### Binary iterations, Fibonacci numbers and permutation of natural numbers

Let $\operatorname{wt}(n)$ be A000120, i.e. the number of $1$'s in binary expansion of $n$ (or the binary weight of $n$).
Also let's consider
$$\ell(n)=\left\lfloor\log_{2} n\right\rfloor$$
and
$$T(n,...

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### What are the hidden assumptions behind Harvey Friedman's claim, CSR?

I'm doing some archeology and trying to understand a claim. As summed up by David Roberts, on the FOM list in 2011:
Let the statement "every infinite sequence of rationals in [0,1] has an ...

1
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1
answer

176
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### Tiling a square with similar non-congruent rectangles. What is the aspect ratio of the rectangles as n grows large?

I recently saw a question here on mathoverflow: «For what n and t can a square be partitioned into n similar rectangles in t congruence classes?», where Joseph Gordon gave a proof that, indeed, a ...

4
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1
answer

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### Why do convoluted convolved Fibonacci numbers pop up from this triangle?

Start with this triangle (OEIS A118981). This triangle is simple to generate with the following recurrence relation (though $T(0,0)$ ends up different from the OEIS version):
$$
T(0,0) = 2;T(1,0) = 1;...

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### My Fibonacci Formula (with combinatorics) [closed]

I'm a high school student, and was playing around with pascals triangle. and ended up taking (weird) diagonals. And I saw Fibonacci numbers, from the sum of the diagonals.
Pascall's triangle is just ...

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4
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### Non-enumerative proof that, in average, less than 50% of tiles in domino tiling of 2-by-n rectangle are vertical?

Is there a non-enumerative proof that, in average, less than 50% of tiles in domino tiling of 2-by-n rectangle are vertical?
It is a nice exercise with rational generating functions (or equivalently, ...

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1
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### Possible small mistake in Bilu-Hanrot-Voutier paper on primitive divisors of Lehmer sequences (?)

I think that I might have spotted I small mistake (a missing $5$-defective Lehmer pair) in the classification of terms of Lehmer sequences without primitive divisors given in:
1 Bilu, Hanrot, and ...

3
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0
answers

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### Is there a closed form of $ \displaystyle \sum_{k=0}^{\infty}{\frac{\phi^{xk}}{k!_F}}$

where $\phi = \frac{1+\sqrt{5}}{2}$ and $k!_F$ is the fibonorial of $k$, or the product of the first $k$ Fibonacci numbers? My hunch is that, this can be represented as a function in terms of the ...

2
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1
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### Is this case of a generalised partition equivalent to Fibonacci numbers?

Let $k=m+\sum^{m+1}_{j=1} a_j$ such that $a,m,k\in\mathbb{N}$ and $a_1$ or $a_{m+1}\geq 0$ with all other $a\geq1$. Note that we assume natural numbers start from $0$ and we have the restriction that $...

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### A generalization of Vajda's identity [closed]

I discovered the identity below which generalizes Vajda's identity concerning Fibonacci Numbers. The identity states that:
if $F_r$ is the rth Fibonacci number, then
$$F_{n+i+x-z}F_{n+j+y+z}-F_{n+x+y-...

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### Requesting proof of closed form of sum involving Fibonacci and Lucas numbers

$$ \sum_{n=0}^{k+1}\frac{3F_{n+1}-L_{n+1}}{2n!}\frac{(k+1)!}{(k-n+1)!}x^{k-n+1}=(\varphi+x)^k\left(\frac{\sqrt{5}}{5}-\frac{\sqrt{5}-5}{10}x\right)+(\psi+x)^k\left(\frac{\sqrt{5}+5}{10}x-\frac{\sqrt{5}...

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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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1
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125
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### Fibonacci with seeds, modulo $n$

Let $n\in\mathbb{N}$ be an integer with $n>1$. For $x_0, x_1 \in \mathbb{Z}/n\mathbb{Z}$ we define the map $\text{fib}_{n, x_0, x_1}: \mathbb{N} \to \mathbb{Z}/n\mathbb{Z}$ by
$0 \mapsto x_0, 1 \...

2
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### Can all (inverse) trigonometric functions with periodic iterates be characterized?

I wonder whether all (composites of) trigonometric and inverse trigonometric functions with periodic functional iterations can be found. In order to specify what I mean by that, let's introduce some ...

17
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1
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### Complexity of a Fibonacci numbers discrete log variation

In my work I encountered the following
FIBMOD PROBLEM:
Given $k,m$ in binary, decide if there exists $n$ such that
$\, F_n = k \,$ (mod $m$). Here $F_n$ is a Fibonacci number.
This is a variation ...

20
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### Reciprocals of Fibonacci numbers

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

54
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### Nontrivial question about Fibonacci numbers?

I'm looking for a nontrivial, but not super difficult question concerning Fibonacci numbers. It should be at a level suitable for an undergraduate course.
Here is a (not so good) example of the sort ...

28
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4
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### Is 8 the largest cube in the Fibonacci sequence?

Can you prove that 8 is the largest cube in the Fibonacci sequence?