# Questions tagged [recurrences]

The recurrences tag has no usage guidance.

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### Existence of limit in a recurrence equation: $\alpha_{n+2}=\frac{\alpha_{n}+(1/\alpha_{n+1})}{2}$ [migrated]

Let be $\boldsymbol{\alpha_{n+2}=\frac{\alpha_{n}+(1/\alpha_{n+1})}{2}}$ a recurrence equation with known $\alpha_0$ and $\alpha_1$. How do you prove that $\lim_{n\to\infty}\alpha_n$ exists? Note that ...

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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:
$f(0) = 4$
$f(1) = 14$
$f(2) = 194$
$f(x+1) = f(x)^2 - 2$

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### Solving Linear Matrix Recurrences

Question:
Are there standard techniques available for solving the following kind of linear matrix recurrence relations:
$$M_1,\cdots,M_k\ \in\ \mathbb{R}^{m\times n}$$
$$ A_1,\cdots,A_k\ \...

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### Sum of products of binomials satisfies recurrence relation?

i need to know if the sequence $(a_n)_{n \geq 0}$ defined by $$a_n =\sum_{s=3n}^{16n} C_{15n}^{s-3n} \;
C_{14n}^{s-2n} \; C_{12n+s}^{12n} $$ satisfies a recurrence relation ( type sequence Apery) or ...

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### Does this deceptively simple nonlinear recurrence relation have a closed form solution?

Given the base case $a_0 = 1$, does $a_n = a_{n-1} + \frac{1}{\left\lfloor{a_{n-1}}\right \rfloor}$ have a closed form solution? The sequence itself is divergent and simply goes {$1, 2, 2+\frac{1}{2}, ...

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### Solution of a 2D Recurrence sequence

Can we solve the following recurrence relation:
$$a_{m,n} = 1 + \frac{a_{m,n-1}+a_{m-1,n}}{2}$$
with $a_{0,n}=a_{m,0}=0$? If not, can we get an estimate of the growth of $a_{m,n}?$
I encountered this ...

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### A 2nd order recursion with polynomial coefficients

I'm hoping to find "exact" an solution to the following simple recursion:
$q_m(j+1) = m \cdot q_m(j) + j(j+m)\cdot q_m(j-1)$
with initial data $q_m(0) = 1$, $q_m(1)=m$, where $m \geq 0$ is an ...

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### If $p_n(a,b)$ is a rational number (or integer) for 3 consecutive values of $n$ then every $p_n(a,b)$ is

Let $a$ and $b$ be two real numbers and $p_n(x,y)$ the polynomial: $$p_n(x,y)=\sum_{i=0}^{n-1}x^{n-1-i}y^{i},$$ where $n$ is a positive integer.
In a previous post I asked if $p_n(a,b)$ was a ...

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### Enumeration of lattice paths of a specific type

One of the approaches to "Special" meanders led (in particular) to the following question:
What is the number $a_{m,n}(\ell)$ of $\ell$-step paths from $(1,1)$ to $(m,n)$ using the ...

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### A conjectural formula for the “minimal degree function”, $k\rightarrow d\min(k)$, attached to recursion, $f\rightarrow A(f)$, in char $3$

THE RECURSION: $f\rightarrow A(f)$
$A: t(\mathbb{Z}/3)[t^3]\rightarrow t(\mathbb{Z}/3)[t^3]$ is the $\mathbb{Z}/3$-linear map with $A(t)=0, A(t^4)=t, A(t^7)=t^4$, and $A((t^9)f)=(2t^9)A(f)+(t^3)A((t^...

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240 views

### $p_n(x,y)=\sum_{i=0}^{n-1}x^{n-1-i}y^{i}$ is always an integer

Does anyone know if the following problem has ever been studied?
Let $a$ and $b$ be two real numbers and consider the polynomial: $$p_n(x,y)=\sum_{i=0}^{n-1}x^{n-1-i}y^{i}$$
where $n$ is a ...

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### Reference requested about the rank of appearance in the Lucas sequences

Given $u_n$ a Lucas sequence, we define the rank of appearance of $m$ in $\{u_n\}_{n\geq 0}$, indicated with $z_u(m)$, as the smallest positive integer $n$ such that $m$ divides $u_n$.
I would like ...

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### Solving 2-dimensional recurrence matrix of homogenous polynomials

I know this isn't research-level mathematics, but I posted this on stack exchange, even offered a bounty but got no responses and no comments, so I am hoping to get an answer here.
In $2012$, ...

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63 views

### Curious if this chaotic recurrence relation has a name and/or interesting properties

I was playing around with 2-dimensional recurrence relations and I stumbled upon this:
$$
x_{n+1} = \sin(k(y_n + x_n))\\
y_{n+1} = \sin(k(y_n - x_n))
$$
I was wondering if this was a known ...