# Questions tagged [recurrences]

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303
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### Sequences that sum up to possible generalization of Euler or up/down numbers (A000111)

Let $a(n,m,k)$ be an integer sequence with e.g.f.
$$A(x)=\operatorname{exp}\left(x + m\int\int (A(x))^k \, dx \, dx\right)$$
I don't know much about integrals, so here's a concrete example:
$a(n,1,3)$...

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177
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### Using Ramanujan-type "Legendrian" sequences to find new formulas for $\frac1{\pi}$?

I. Recurrences
(Continued from this post.) In Cooper's 2012 paper, "Sporadic sequences, modular forms and new series for 1/π", he did a computer search for the recurrence relation,
$$(n+1)^3 ...

4
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1
answer

126
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### On four Ramanujan-type "Legendrian" sequences with a 3-term recurrence?

I. Recurrences
In a previous post, it was mentioned how Almkvist-Zudilin did a computer search for solutions to the recurrence relation,
$$(n+1)^3s_{n+1}=(2n+1)(an^2+an+b)s_n+c\,n^3s_{n-1}$$
within a ...

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46
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### Application of the series reversion

Let $f(n)$ be an arbitrary function such that $f(n)\in\mathbb{Z}$.
Let $a(n)$ be an arbitrary integer sequence such that $a(0)=1$.
Let $b(n)$ be an integer sequence such that
$$b(2^m(2n+1))=\sum\...

5
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349
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### On H. Cohen's four continued fractions for $\zeta(3), \zeta(5), \zeta(7)$?

After 6 years from this old MO post, I finally find in the literature polynomials of deg-$5$ for the continued fraction of $\zeta(5)$.
I. Recurrences involving $\zeta(5)$
In Cohen's 2022 paper, ...

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78
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### Simplification of summation and reverse search

Let $f(n)$ be an arbitrary function such that $f(n)\in\mathbb{Z}$.
Let $b(n)$ be an integer sequence such that
$$b(2^m(2n+1))=\sum\limits_{k=0}^{m}f(m-k)b(2^kn), b(0)=1$$
Let $s(n,m)$ be an integer ...

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2
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525
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### On 12 cfracs: for Catalan's $K$, Gieseking's $\kappa$, and $\pi^2$, $\pi^3$, plus three for $\zeta(3)$ using Zagier's "six sporadic sequences"

I. Some functions
As these will be used in the continued fraction evaluations below, recall the Riemann zeta function $\zeta(s),$ and Dirichlet beta function $\beta(s),$
$$\beta(s) = \sum_{n=1}^\infty\...

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### Sequences that sum up to $\frac{m^{n+1}}{n+1}\binom{2n}{n}{}_2F_1(1,n+\frac{1}{2}; n+2; -4m(m-1))$

Let $a(n,m)$ be an integer sequence such that
$$a(n,m)=\frac{m^{n+1}}{n+1}\binom{2n}{n}{}_2F_1(1,n+\frac{1}{2}; n+2; -4m(m-1))$$
Let
$$\ell(n)=\left\lfloor\log_2 n\right\rfloor$$
$$f(n)=n-2^{\ell(n)}$$...

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2
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181
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### Finding a new level-$7$ pi formula using the relation $j_{7A}(\tau) = \left(\sqrt{j_{7B}(\tau)}+\frac{7}{\sqrt{j_{7B}(\tau)}}\right)^2$

(Note: This third method continues from this post.)
There are level-$7$ pi formulas based on the McKay-Thompson series $T_{7A}$ and Cooper's $s_7$ sequence in this paper. This third method, among ...

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### $\frac{m(m+k+1)^n+k}{m+k}$ as closed form for subsequence of the partial sums

Let $a(n,m,k)$ be an integer sequence such that
$$a(n,m,k)=\frac{m(m+k+1)^n+k}{m+k}$$
There are many sequences in the OEIS that are special cases of a given sequence family:
$a(n,1,1)$ - A007051
$a(n,...

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47
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### Counting problem, tiling rectangle with two types right isosceles triangle

How many ways are there to tile a rectangle of size $m\times n$ with two types of isosceles triangle, type 1 having area $\frac{1}{2}$ and type 2 having area 1?
I know with only type 1 there are $2^{...

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49
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### Existence of integer sequence under simultaneous constraints

Does there exist a function $f:\Bbb N\to\Bbb N$ such that \begin{align}a_{n+1}&=f(a_n)\\a_{f(n)+1}&=a_n\end{align} implies $\{a_n\}_{n\ge0}$ is a non-constant, positive integer sequence? ...

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### Representation theorem for multivariate homogeneous linear recurrences on Z^d?

Let $f:\mathbb{Z}^d \to \mathbb{C}$ satisfy a homogeneous linear recurrence for some coefficients $a_\Delta \in \mathbb{C}$,
$$\forall x \in \mathbb{Z}^d. \quad \sum_{\Delta \in B_k(0)}a_\Delta f(x+\...

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99
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### Solve the recurrence relation with 2 variables

We have the following recurrence relation:
\begin{equation}
f(n,m) = f(n-1,m) g_{\alpha, \gamma}(n,m) + f(n,m-1) g_{\beta, \gamma}(n,m) \\
g_{\alpha, \gamma}(n,m)= \sum^{n}_{i = 0} \sum^{m}_{j = 0} \...

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### Product as closed form for subsequence of the partial sums

Let $a(n,m,k)$ be an integer sequence such that
$$a(n,m,k)=\prod\limits_{q=0}^{n-1}\sum\limits_{i=0}^{m-1}\sum\limits_{j=0}^{m-i-1}\binom{i+j-1}{j}k^{i+j}q^i$$
Let
$$\ell(n,m)=\left\lfloor\log_m n\...

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### Generalization of Lucas sequences to order 3 (and above)

For fixed integer parameters $(P,Q)$, Lucas sequences represent a pair of complimentary integer sequences satisfying the same recurrence with the characteristic polynomial $f(x):=x^2 - Px + Q$. The ...

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### On a generalization of A113227 as a subsequence of the partial sums

This question is just a generalization of the one of my previous questions.
Let
$$a(n,m,k)=\sum\limits_{i=1}^{n}u(n,m,k,i)$$
where
$$u(n,m,k,i)=u(n-1,m,k,i-1)+(m-1)(i+k-1)\sum\limits_{j=i}^{n-1}u(n-1,...

6
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### How to solve recurrence relation with 2 variables？

I have the following recurrence relation and boundary condition?
$$
f(n,m) = \frac{\alpha n}{n+m} f(n-1,m) + \frac{\beta m}{n+m} f(n,m-1) + 1
$$
$$
f(n,0) = \frac{1-\alpha^{n+1}}{1-\alpha}, f(0,m) = \...

2
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0
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### Coefficient growth upper bound of a recurrence relation

Consider the recurrence relation one can obtain from the radial Schrödinger equation for the hydrogen atom, where $\psi(r)=\sum_{n=0}^{\infty} a_nr^n$:
$$(n+3)(n+2)a_{n+2}+2a_{n+1}=2|E|a_n, n\geq0$$
...

3
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0
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### Closed form for subsequence of the partial sums of generalized A329369

Let $a(n,m,k)$ be an integer sequence such that
$$a(n,m,k)=\sum\limits_{i=0}^{n}{n\brace i}(m-k)^{n-i}\prod\limits_{j=0}^{i-1}(kj+1)$$
Here ${n\brace k}$ is the Stirling number of the second kind.
...

3
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109
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### Sequences that sum up to Dowling numbers

Let $a(n,k)$ be the sequence of $k$-Dowling numbers (for more information see A007405 and its CROSSREFS section) with e.g.f.
$$\operatorname{exp}\left(x + \frac{\operatorname{exp}(kx) - 1}{k}\right)$$
...

2
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0
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### Sequences that sum up to the many sequences in the OEIS

Let
$$a(n,m,k)=\frac{1}{n}\sum\limits_{j=0}^{n}[n+kj\geqslant 0]\binom{n}{j}\binom{n+kj}{j-1}(m-1)^{j-1}$$
Here square brackets denote Iverson brackets.
There are many sequences in the OEIS that are ...

6
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1
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159
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### Sequence that sums up to the number of permutations avoiding the pattern $1-23-4$

Let $a(n)$ be A113227, i.e., the number of permutations on $[n]\equiv \{1, \ldots, n\}$ avoiding the pattern $1-23-4$.
The sequence begins with
$$1, 1, 2, 6, 23, 105, 549, 3207, 20577, 143239, 1071704,...

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### Recurrence for the number of permutations with a given excedance set

Let
$$\ell(n)=\left\lfloor\log_2 n\right\rfloor$$
$$f(n)=n-2^{\ell(n)}$$
$$T(n,k)=\left\lfloor\frac{n}{2^k}\right\rfloor\operatorname{mod}2$$
$$\operatorname{wt}(2n+1)=\operatorname{wt}(n)+1, \...

1
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0
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130
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### Recurrence for the A284005

Let
$$\ell(n)=\left\lfloor\log_2 n\right\rfloor$$
$$f(n)=n-2^{\ell(n)}$$
$$T(n,k)=\left\lfloor\frac{n}{2^k}\right\rfloor\operatorname{mod}2$$
$$\operatorname{wt}(2n+1)=\operatorname{wt}(n)+1, \...

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1
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189
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### Polynomial solutions to a difference equation

This question may look unmotivated, but is connected with continued fractions for $\pi^2$.
Let $n$ be a nonnegative integer, and consider the difference equation
$$(x+2n+4)(x+n+1)P(x+1)-(x-1)(x+n-1)P(...

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0
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### Urn model with delayed replacement

Suppose I have x red and y blue balls. At each timestep I draw a ball with probability $$P(\text{red ball}) = (x/(x+y))^z, P(\text{blue ball}) = 1-P(\text{red ball})$$ where z is fixed.
Each ball is ...

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0
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78
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### Permutation to get Stolarsky representation from lazy Fibonacci (dual Zeckendorf) representation

Let $a_1(n)$ be A200714, i.e., Stolarsky representation interpreted as binary to decimal integers. The sequence begins with
$$0, 1, 3, 2, 7, 5, 6, 15, 4, 11, 13, 14, 31, 10, 9, 23, 12, 27, 29$$
Let $...

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0
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### Methods for holonomic recurrences

I wanted to ask if anyone knows of good texts/resources on methods for solving holonomic recurrence relations (if there are any general analytical approaches):
$$p_1(n)a(n)+p_2(n)a(n-1)+\dotsb+p_k(n)a(...

1
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0
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### The limit set of consecutive applications of linear transforms to the single segment

Problem. Consider $n$ positive integers $1 < a_1\le \ldots \le a_n$ and $I = \left[\frac{1}{a_n - 1}, \frac{1}{a_1 - 1}\right]$. For each $a_k$ define the linear transform $\phi_k\colon x\mapsto \...

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### Limit of recursion relation

Consider the sequence of functions $\{F_n\}_{n\in \mathbb{N}}$, where each $F_n$ is defined on $\{0,...,n\}$ by recurrence of the following form: $$ F_n(0)=3 \textrm{
and }F_n(k)=\frac{1}{k^2}+\frac{\...

0
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0
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### Families of 1st order recursive relations with closed forms

A general 1st order recurrence relation can be expressed as $X_{n+1} = f(X_{n},n)$.
I am looking for a source, list of functions or statement about $f$ such that an $F$ exists which expresses $X_{n} = ...

0
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0
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### Examples of $f(n,m)$-solvable recurrences

Let $a_k(n)$ be an integer sequence given by recurrence relation
$$a_k(2n+1)=a_k(n), a_k(2n)=a_k(n)+\sum\limits_{j=1}^{i}a_k(g_j(n)), a_k(0)=1$$
Here $g_j(n)$ are some functions in most cases based on ...

3
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1
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194
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### Powers of $2$ up to $2^{m-1}$ from a polynomial of degree $m-1$

Let $T(n,k)$ be a triangle of coefficients such that $T(n,k)\geqslant0$ for $n>0$, $0<k\leqslant n$, $0$ otherwise. Also
$$T(2n+1,1)=\frac{1}{2n+1}, T(2n,1)=0$$
$$T(n,k)=\frac{1}{n}(T(n-1,k-1)+(...

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### Identifying terms of a linear recurrence sequence using a congruence

Let $(n_i)_{i=0}^\infty$ be a sequence of integers satisfying a linear recurrence with integer coefficients,
$$ n_{i+\ell} = \sum_{k=0}^{\ell-1} a_k n_{i+k},$$
and for simplicity assume that $a_0 = \...

2
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0
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### Closed form for coefficients related to excedance set of permutation

Working on suitable closed form for A329369, I discovered very useful coefficients, which have the following recurrence relation:
$$T(0,1)=T(0,2)=1$$
$$T(n,1)=1, n>0$$
$$T(0,k)=0, k>2$$
$$T(2n+1,...

0
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0
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100
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### One variable recurrence relation and two variable recurrence relation

Let $q(n)$ be A007814, i.e., number of trailing zeros in the binary representation of $n$. Here
$$q(2n+1)=0, q(2n)=q(n)+1$$
Let $a(n)$ be A329369. Here
$$a(2n+1)=a(n), a(2n)=a(n)+a(n-2^{q(n)})+a(2n-2^{...

0
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1
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89
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### Recurrence for the number of steps required to get one ball in each box

Given $n$ balls, all of which are initially in the first of $n$ numbered boxes, $a(n)$ is the number of steps required to get one ball in each box when a step consists of moving to the next box every ...

0
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0
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124
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### How many steps does this subtractive recurrence take?

Given $\alpha\in(0,1)$ and $c\geq1$.
$n$ here is in naturals $\mathbb N$.
$$T_0=n$$
$$T_i=T_{i-1}-\frac{\lfloor{T_{i-1}}^\alpha\rfloor}c\mbox{ at every }i\in\mathbb N$$
is the recursion.
At what $i$ ...

1
vote

1
answer

99
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### Number of steps required to get one ball in each box for $n=2^k$

Given $n$ balls, all of which are initially in the first of $n$ numbered boxes, $a(n)$ is the number of steps required to get one ball in each box when a step consists of moving to the next box every ...

4
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2
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358
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### Is there a recurrence for the coefficients of the Laurent series expansion of $\frac{1}{1-e^{e^x - 1}}$?

So there's an elementary (but in my opinion quite interesting!) result which is that the Laurent series expansion of
$$\frac{1}{1-e^x} = -\frac{1}{x} + \frac{1}{2} - \frac{1}{12}x - \frac{1}{720}x^3 \...

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0
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### Recurrence for permutation of A007306 (denominators of Farey tree fractions)

Let $a(n)$ be A071585, i.e., numerator of the continued fraction expansion whose terms are the first-order differences of exponents in the binary representation of $4n$, with the exponents of $2$ ...

3
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1
answer

139
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### Using generating functions to construct or solve differential equations

I know that $T_n(x)$ is the solution of the differential equation $(1-x^2)y''-xy'+n^2y=0$, where
$$
T_n(x)=\begin{cases}
T_n(x)=1 & \text{if $n=0$}\\
T_n(x)=x & \text{if $n=1$}\\
T_{n}(x)=...

0
votes

1
answer

60
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### Vector recurrences (asymptotic property)

Fix $m\in \mathbb{N}.$
For each $n\in \mathbb{N},$ let $A_n\in \mathbb{M}_{m}(\mathbb{C}),$ $X_n\in \mathbb{C}^m,$ and $B_n\in \mathbb{C}^m.$ Suppose that
$$X_{n+1}=A_n X_n+B_n,$$
$$\lim_{n\rightarrow ...

6
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0
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150
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### 3-term recurrence relation including integral or differential operator for polynomials

Sequences of polynomials with a 3-term recurrence relations are well known for orthogonal polynomials. Do recurrence relations using differential or integral operators also appear in some theories?
I ...

14
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0
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240
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### A conjectured rational generating function

In regard to my question here, let $G_n$
be a sequence of positive integers satisfying
$\lim_{n\to\infty}G_n=\infty$, such that the generating function
$\sum_{n\geq 1} G_nx^n$ is rational. Let
$$ P_n(...

0
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0
answers

90
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### Results regarding families of recursive polynomials

I'm interested in the properties of particular families of recursively-generated, complex polynomials. In particular, let $p_0(x) = e^{i \phi_0}$ and $p_1(x) = e^{i [\phi_1 - \phi_0]} x$. Then, for $n ...

5
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0
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134
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### Behavior of $n\sqrt{3}-a_n\sqrt{n}$ as $n\to\infty$, where $a_{n+1}=a_n+n/(a_1+\dots+a_n)$

For a fixed real $a>0$, define the sequence $(a_n)_{n>0}$ by $a_1=a$ and $$a_{n+1}=a_n+\frac{n}{a_1+\dots+a_n}.\qquad(n>0)$$
It is known (see this answer) that $$\lim_{n\to\infty}\frac{a_n}{\...

1
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0
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67
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### Recurrences (based on Fibonacci numbers) for the first differences of numbers filtred by equality of binary functions

First we need to set some binary functions:
Let
$$\ell(n)=\left\lfloor\log_2 n\right\rfloor$$
Let $\operatorname{wt}(n)$ be A000120, i.e., $1$'s-counting sequence: number of $1$'s in binary expansion ...

1
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0
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92
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### Recurrence relation of the form R(x,y)=yR(x-1,y)+(x-(y-1))R(x,y-1)

Consider the recurrence
$$
R(x,y)= yR(x-1,y)+ (x-(y-1))R(x,y-1)
$$
where for any $R(p,c)$, $c$ does not exceed $p$, and $R(p,p)=R(p,1)=1$.
I´ve tried to write $R(x,y)$ as a sum of coefficients of $R(...