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### Solving a two dimensional non-homogenous linear recurrence

How one can solve the following recurrence:
\begin{align}
X[i,0] &=0 \quad \forall i =1,\ldots, m\\
X[m,n] &= a_n X[m,n-1]+b_n \sum_{i=k_m}^{m-1}X[i,i] +c_n
\end{align}
Where $a_i\ge 1 ,~ 0 ...

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### Asymptotic growth of recurrence relation $x_n=\min\limits_{n_1+n_2=n}(a(x_{n_1}+x_{n_2})+2n_1n_2)$

Suppose $a>0$ . Define
$$x_n=\min_{n_1+n_2=n}(a(x_{n_1}+x_{n_2})+2n_1n_2) \text{ with } x_1=0.$$
Can we find $r>0$ such that there exists two positive constant $c_1,c_2$ such that
$$c_1<\...

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### Multiplying by irrational numbers in combinatorial problems

This is getting no attention on stackexchange.
Everybody knows that the number of derangements of a set of size $n$ is the nearest integer to $n!/e$.
It had escaped my attention until last week, ...

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### Beyond Collatz: A $5n+1$ conjecture? [closed]

Let
$$x_{n+1} = \begin{cases} x_n/2 &;\text{if } x_n \equiv 0 \pmod{2}\\ k\,x_n+1 &; \text{if } x_n\equiv 1 \pmod{2} \end{cases}$$
and $k=3$ and $x_n\in\Bbb N$. Collatz conjectured for this ...

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185 views

### Non-crystallographic cluster algebras

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. Here, a seed $(\mathbf{...

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### Approximate closed-form solution for a recurrence

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}^{\min(b,e)}{e\choose i}{(...

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**1**answer

338 views

### Closed form solution to an iterative equation.

$f(n+1) = f(n) + f(n)^{a}$ 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 to interpolate ...

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338 views

### Giving a general term of a recursive function, and upper bound for it

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 $l_{t+1} = l_t$ otherwise
Let $b_{t+1}...

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### References on techniques for solving equations with discontinuous functions such as floor and ceiling?

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 asking for some good ...

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### Master theorem for probabilistically inspired recurrences

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,k-1) + p_4 f(i-1,j-1,...

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### Linear Recurrence Relations in 2 Variables with Variable Coefficients

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 of solutions to this ...

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### Methods for solving two variable recurrence

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 system is defined for $...

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368 views

### Books request on nonlinear recurrence relations.

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 to differential ...

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### A recursive Double sequence related to uniform Cardinal B-spline

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

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### Trying to get an idea of the maths I could use for this optimization problem

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 the concepts in this ...

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### Perturbation analysis for three term recurrences

Jacobi polynomials, denoted by $J^{(\alpha,\beta)}_n$, on $[-1,1]$ satisfy a three term recurrence
$$ J_{n+1}^{(\alpha,\beta)}(x) = (A_n+B_nx)J^{(\alpha,\beta)}_n + C_nJ_{n-1}^{(\alpha,\beta)}(x), \...

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### Partitions of central sets via dynamical systems

In the book "Recurrence in Ergodic Theory and Combinatorial Number Theory", 1981, Furstenberg introduced the notion of central sets.
He proved in Theorem 8.8 that in each finite partition of $\mathbb{...

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### Uniqueness of values in recurrence relations

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,$ $x_1 = b - 1,$ $x_n = ...

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701 views

### A question on the Laurent phenomenon

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) with complicated
...

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### How to solve a specific multivariate recurrence relation (or general ones)

How do you solve this recurrence (or multivariate recurrences in general)? Note that $p\in[0,1]$ and $n\in\mathbb{N}$ are given constants, where $np\leq 1$.
$$f:(\mathbb{N}\cup\{0\})\times(\mathbb{N}\...

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**1**answer

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### Three term recurrence relation.

For given $n,\ell\in\mathbb N_0$, I am interested in studying the following recursion relation for some $\mu\in\mathbb R$:
$$\sqrt{-1} \tfrac{j(\ell-j+1)(n-j+1)(n+j+1)}{2(2j-1)(2j+1)} a_{j-1} - \...

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### On bounding the average cost of top-down merge sort

Let $A_n$ be the average number of comparisons to sort $n$ keys by merging them in a top-down fashion (see any algorithm textbook). It can he shown that
$$
A_0 = A_1 = 0;\quad A_n = A_{\lfloor{n/2}\...

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### Recurrence T(N)=T(N/LOGN)+1 [closed]

I'm stuck on the following reccurrence :
T(N)=T(N/LOGN)+1,IF N>2
T(N)=0,if 0<=N<=2
I need a function T(N) for all N>0
Is there some method for solving ...