# Questions tagged [recurrences]

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### Is there a rectangular tiling based on the Padovan sequence? [closed]

I'm thinking of developing a rectangular tiling based on the Padovan sequence (think of the Fibonacci mosaic). It seems like something that should exist, but I can't find anything in the literature. ...
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### Find the closed-form expression for $c_n$ of this recursive sequence [closed]

$$c_{n + 1} = 2\cdot|c_n| - \sqrt{c_n^2 + 16}$$ with $c_0 = 3$. My question on math stackexchange was closed because lack of details. Let me clarify: I'm an amateur novelist, I don't know anything ...
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### exponential growth of random fibonacci sequences

I don't know if this is the right place to ask this question because I have the impression that this is quite an elementary one.. But I thought, maybe someone here already read this paper (or would be ...
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### Convergence with the recurrence $T_{n+1}=T_n^2-T_n+\frac{n}{p_n}$

For each integer $n\geq 1$ I define the recurrence $$T_{n+1}=T_n^2-T_n+\frac{n}{p_n},$$ with $T_1=1$, where $p_k$ denotes the $k-th$ prime. So multiplying by $(-1)^n$ and telecoping gives that for ...
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### Prove that the Mertens function is invariant under matrix inversion for $n>3$

Consider the lower triangular matrix $T$ with the definition: $n \leq3 :$ $$T(n, 1)=1$$ $n>3:$ $$T(n,1)=x_n$$ $n \geq k:$ $n \leq 3:$ $$T(n,k)=1$$ $n \geq k:$ $n>3:$ $k=2 \text{ or } k=3:$ ...
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### Two conjectural identities involving $\zeta(3)$ and the golden ratio $\phi$

Let $\phi$ be the famous golden ratio $\frac{\sqrt5+1}2$, and let $\zeta(3)=\sum_{n=1}^\infty\frac1{n^3}.$ In 2014, in the paper Zhi-Wei Sun, New series for some special values of $L$-functions, ...
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### A 2d recurrence equation representing a step-free continuous-time Markov process on N^2 with frequency-dependent rates

$\textbf{Background:}$ Consider a particle in $\mathbb{N}^2$ starting at a point $(x,y)$ that can only move one step at a time along the lattice. The particle moves each up or down in continuous time ...
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### Solving an recursive sequence [closed]

I have an recursive sequence and want to convert it to an explicit formula. The recursive sequence is: $f(0) = 4$ $f(1) = 14$ $f(2) = 194$ $f(x+1) = f(x)^2 - 2$