Background Fomin and Zelevinsky have introduced cluster algebras in an influential article. To define a cluster algebra, Fomin and Zelevinsky have defined a mutation of seeds. Her …

Question: A codeword is made up of the digits 0,1,2,3. A codeword is defined as legitimate if and only if it has an even number of 0’s. Let an be the number of legitimate codewords …

Find an (approximate) closed-form solution for $S(m, b)$. $$S(m,b)=\sum_{i=0}^{\lfloor (e-1)/2\rfloor}{e \choose i}S(m-1, b-i) \quad + \sum_{i=\lfloor (e-1)/2\rfloor+1}^{\ …

$f(n+1) = f(n) + f(n)^{a}$ where $a \in (0,1)$ and $n \ge 1$ with $f(1) = m$. If $a=0$, we see $f(n) = m + n - 1$ and if $a=1$, we see $f(n) = 2^{n-1}m$. So the recursion seems t …

Here I describe the sort of reference I'm after with a motivating example. I am not seeking solutions to my equations on this forum; I'm quite happy to do that myself. Rather, I'm …

Let a constant $B \ge 1$, and let $l_1 = 0$, $b_1 = 0$ be the values of $l$ and $b$ (respectively) at time $t = 1$. Let $l_{t+1} = l_t + 1$ if $b_i < B$, and $l_{t+1} = l_t$ ot …

I have a recurrence $$f(i,j) = 1+ \frac{N-i}{N} f(i,j-1) + \frac{i}{N} f(i-1, j)$$ $$f(i,0) = 0$$ $$f(0,j) = j$$ I would like to compute $f(N,M)$ in terms of N and M. The sys …

Consider the following recurrence relation: $$-2a_{n,m} +a_{n-1,m}+a_{n,m-1}=0,$$ where $a_{n,m} \in \mathbb{C}.$I would like a purely combinatorial way to understand the subspace …

Is there a general solution for multi-variable recurrences of the following form which come from Markov chain analysis? $f(i,j,k) = p_1 f(i-1, j,k) + p_2 f(i, j-1,k) + p_3 f(i,j, …

Given a sequence $A_n(k)$ defined as follows: $A_0(0)=1$, $A_0(k)=0$ for all nonzero integers $k$ and $$A_{n}(k)=(n+1-k)^2A_{n-1}(k-1)+2(n(n+1)-k^2)A_{n-1}(k)+(n+1+k)^2A_{n-1}(k+1 …

Hi, do you have some sort of a bibliography on advanced techniques in recurrence equations, such as nonlinear ones and others? As I see it recurrence equations are quite similar t …

Given an integer $k > 1$, define the sequences $X(k,n), Y(k,n)$ as follows: $a=4k-2,$ $y_0 = 1,$ $y_1 = a + 1,y_n = ay_{n-1} - y_{n-2}$ $b = 4k + 2,$ $ x_0 = 1,$ $x_1 …

Firstly, apologies if some of the notation or terminology is odd, or if I am defining functions that have standard notation associated with them already - I am not familiar with th …

This question is motivated by my answer to 109955. It gives a recurrence relation satisfied by a function $P(n)$ whose terms a priori are rational functions (of three variables) wi …

I have the following recurrence relation: \begin{equation} A[n]=f_A[n-1] A[n-1] + f_B[n-1]B[n-1], \\ B[n]=g_A[n-1] A[n-1] + g_B[n-1]B[n-1], \end{equation} where $f_{A/B}$ and $g_ …