# 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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### Infinite Fibonacci word generation

I was reading about the Fibonacci sequence and stumbled upon the Fibonacci word. It looks pretty straight forward, but one thing I do not understand. Probably something trivial. According to Wikipedia ...
1 vote
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### Calculate the number of functions $f:\{1;2;\cdots;n\} \to \{0;1;\cdots;n-1\}$

For $n \in \mathbb{N}^*$, calculate the number of functions $f:\{1;2;\cdots;n\} \to \{0;1;\cdots;n-1\}$ such that $f(k) \leq k-1$ for all $k \in \{1;2;\cdots;n\}$, and there do not exist three numbers ...
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### Linear combinations of solutions to linear recurrences

I am working through MIT's Mathematics for Computer Science through OCW. Currently on homogeneous linear recurrences. A link with the course textbook is attached. My question is with the proof for the ...
36 views

### The Fibonacci sequence: ratio of mature to young rabbit pairs within each generation also converges on the Golden Ratio (online or other reference?)

In looking at the breeding rabbit pair model which lies behind the Fibonacci sequence, one observes that for any given population of A(n) adult pairs (breeding) and Y(n) young pairs (non-breeding) in ...
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### Induction Proof for the Sum of the First n Fibonacci Numbers

I'm trying to prove that there exists a formula for the sum of the first n Fibonacci numbers, using induction. $$\sum_{i=1}^{n} F_i$$ For my class, we have denoted the recursive definition of the ...
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1 vote
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### Ratio of Fibonacci numbers

I have observed the following property: If $F_n$ denotes the $n^{th}$ Fibonacci number and $F_1=1; F_2=1$ Then $\frac{F_{2n+1}}{F_{2n}}>\phi$ And $\frac{F_{2n}}{F_{2n-1}}<\phi$ For all natural ...
116 views

### Can the Fibonacci Spiral be expressed as a polar equation?

Related to this question: can the Fibonacci Spiral be expressed as a polar equation? I know the Golden Spiral can be and the Fibonacci differs in that it uses consecutive arcs of a circle for each ...
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### Fibonacci Spiral vs Golden Spiral?

I watched this video on constructing the Fibonacci Spiral. Does it differ from the Golden Spiral? The Fibonacci Spiral is constructed by using arcs of a circle on consecutive squares where the ...
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### Other than $1$, does the sequence $1,12,123,1234,\cdots$ contain a Fibonacci Number?

Other than $1$, does the sequence $1,12,123,1234,\dots$ contain Fibonacci Numbers? This is sequence A007908, and the numbers listed in this sequence is obtained by concatenating the first $n$ ...
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### Why does Lucas get credit for Fibonacci's progression?

A so-called "Lucas Number" is, to me, nothing more than a standard Fibonacci progression of $n_2+n_1$ with a different starting point. There are infinitely many similar progressions, so why ...
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1 vote