# Questions tagged [recurrence-relations]

Questions regarding functions defined recursively, such as the Fibonacci sequence.

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### How to solve the recurrence relation $a_n=\sqrt{a_{n-2}}$

I don't exactly remember where I got this from or when, but I remember seeing this recurrence relation in a Youtube video that was just going over the different types of recurrence relations. I have ...
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### Consider the following recurrence, $a_n=\frac{4a_{n-1}^3+2a_{n-1}-a_{n-2}}{1+4a_{n-1}a_{n-2}}$ where $a_0=0, a_1=1$. Show every $a_n$ is an integer.

Consider the following recurrence, $$a_n=\frac{4a_{n-1}^3+2a_{n-1}-a_{n-2}}{1+4a_{n-1}a_{n-2}}$$ where $a_0=0, a_1=1$. (a) Show that every $a_n$ is an integer. (b) Find the general term of $a_n$. What ...
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### Prove that if $\forall n\in\Bbb N\quad a_{n+1}^2-a_{n+1}=a_n\in\Bbb Q$, then the sequence is constant

This question was posted, downvoted and closed today (2022 Thailand Olympiad problem) and 8 days ago ($f(x+1)^{2} - f(x+1) = f(x)$. What values of $f(1)$ allow $f(x)$ to be always rational if $x$ is ...
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### Recurrence relation for integer sequences raised to the power of $n$

An interesting pattern can be observed when considering a sequence of positive integers raised to some power $n$. When a sequence of continuous integers $i$ (of length >=n) is raised to the power ...
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### At every step, add $k$ previous terms, then remove all zeros. Then $k=3$ leads to a $300,056,874$-cycle.

Recently, I've been interested with the sequence of the form: $$a_n =\text{Zr}(\sum_{j=1}^k a_{n-j}),\ \ n≥k$$ $$a_n=1,\ \ 0≤n<k$$ Where $\text{Zr}(x)$ is just $x$ with $0$ removed in its digits (i....
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### Solving recurrence relation to find a generating function

Say I have the recurrence: $a_t(u,v) = (a_t(u-1,v-1) + a_t(u-1,v)) q^{u-2v+t}$ with initial values: $a_t(u,v) = 0$ for $u < 0$ , $v < 0$, or $v > L$. Where $L = \lfloor(u-t-1)/2\rfloor$ ...
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### Sequences of the form $A(n) = A(A(n-1)\bmod n)^2$

$$A(0)= x \in\mathbb{Z}^+,\ A(n) = A(A(n-1) \bmod n)^2$$ At first glance, one would think that such sequence would grow very fast. But my testing suggest that this sequence actually ends with $x^4$ ...
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### find the coefficient $a_n$ of a power series

I have the power series $$f(z)=\frac{2}{(1-3z)^{\frac{2}{3}}}+e^{\frac{1}{2}z^2},$$ and I am supposed to find an explicit expression for the coefficient of the corresponding sum representation of ...
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