All Questions
Tagged with integer-sequences co.combinatorics
160 questions
5
votes
2
answers
393
views
What is this sequence counting?
While solving (a system of) a system of linear equations level-by-level recursively, I am finding some redundant equations for level $n\geq5$. The reason why the redundancies arise is because $P(n)\...
- 153
5
votes
1
answer
179
views
A common combinatorial description for a certain type of recurrences
For integer-valued sequences $(x_n)_{n=0}^\infty$, consider recurrences of the form
$$x_n=ax_{n-1}+(bn+c)x_{n-2} \tag{$*$}\label{star}$$
for $n\ge2$, where $a,b,c$ are integers.
There seem to be many ...
- 128k
5
votes
1
answer
345
views
Why does this "factorial sequence" appear in the OEIS?
For a reciprocal of a polynomial, $f = \frac{1}{p}$, we (presumably) may construct a sequence $(c_n)_{n=0}^\infty$ such that for all $N\ge 0$
$$f(k)k! = \sum_{n=0}^{N-1} c_n(k-n)! + O((k-N)!). $$
I ...
- 3,499
5
votes
2
answers
1k
views
Proof that $3^ns + \sum_{k=0}^{n-1} 3^{n-k-1}2^{a_k}=2^m.$
How would I go about proving the following:
For any odd positive integer $s$, there exists a sequence of nonnegative integers $( a_0, a_1, \cdots, a_{n-1})$ and a nonnegative integer $m$ such that,
$...
5
votes
1
answer
303
views
Simply generated sequences with mysterious differences
Suppose that $a_0 < a_1,$ $b_0 < b_1,$ and $$a_n=a_1b_{n-1}+a_0b_{n-2}+qn+r$$ for $n \geq 2$, where $a_0,a_1,b_0,b_1,q,r$ are integers such that $(a_n)$ and $(b_n)$ are increasing and ${(|a_n|)}$...
- 3,443
5
votes
1
answer
310
views
In the Oldenburger-Kolakoski sequence, is #1s = #2s infinitely many times?
The Oldenburger-Kolakoski sequence, $OK$, is the unique sequence of $1$s and $2$s that starts with $1$ and is its own runlength sequence:
$$OK = (1,2,2,1,1,2,1,2,2,1,2,2,1,1,2,1,1,2,2,1,2,1,1,\ldots).$...
- 479
5
votes
1
answer
384
views
Flow of an integer
I've stumbled across this family of flow networks, and posted the sequence of maximal flows to OEIS: A238729. I can't find any reference to it either. Has anyone seen it?
Here is the description:
...
5
votes
0
answers
183
views
On the polynomials $\sum_{k=0}^n\binom{n+k}k^m q^k$
A sequence of polynomials
$$P_0(q),\ P_1(q),\ P_2(q),\ \ldots$$
with real coefficients is called $q$-log-convex if for each $n=1,2,3,\ldots$ every coefficient of the polynomial $P_{n+1}(q)P_{n-1}(q)-...
- 15.6k
5
votes
0
answers
307
views
On $s$-additive sequences
For a non-negative integer $s$, a strictly increasing sequence of positive integers $\{a_n\}$ is called $s$-additive if for $n>2s$, $a_n$ is the least integer exceeding $a_{n-1}$ which has ...
- 791
5
votes
0
answers
133
views
Formula and smallest solution for the A260711
Let $a(n)$ be A260711 without initial $0$ (i.e., numbers of the form $x^2 - y^2$ with $x > y$ where $x$ and $y$ are odd, $x + y$ is a power of $2$).
The sequence begins with
$$
8, 16, 32, 48, 64, ...
- 4,948
5
votes
0
answers
1k
views
A generalization of the difference of squares identity
Let us find explicit integer functions for the coefficients of the monomial expansion of
$$
Q \left( x_1, \ldots , x_n \right) = \prod_{\left( \kappa_1, \ldots , \kappa_{n-1} \right) \in \{-1,1\}^{n-1}...
- 149
4
votes
2
answers
240
views
Databases for sequences indexed by partitions
Is there a database for sequences indexed by partitions similar to Sloane's OEIS? I mean, I am aware that in the OEIS there are some arrays indexed by partitions, but I feel as though most of such ...
- 16.9k
4
votes
1
answer
120
views
Avoiding equality of partial sums of two different aperiodic sequences
Consider two distinct sequences of positive integers, $a_{n}|_{n=1}^{\infty}$, and $b_{n}|_{n=1}^{\infty}$ such that for either sequence no period exists. The elements of both sequences are drawn from ...
- 163
4
votes
1
answer
245
views
Count weighted integer compositions
What is the asymptotic growth of the sequence
$$a_n:=\sum_{k\geq 0} 3^k c_{n,k},$$
as $n\rightarrow\infty$, where $c_{n,k}$ denotes the number of integer compositions of $n$ with exactly $k$ many 2s?
...
- 499
4
votes
1
answer
148
views
Closed form for the A110501 (unsigned Genocchi numbers (of first kind) of even index)
Let $a(n)$ be A110501 (i.e., unsigned Genocchi numbers (of first kind) of even index). Here
$$
a(n) = \sum\limits_{i=1}^{\left\lfloor\frac{n}{2}\right\rfloor}\binom{n}{2i}a(n-i)(-1)^{i-1}, \\
a(1) = 1
...
- 4,948
4
votes
1
answer
322
views
Combinatorics related plane geometry
There are $n$ men, standing one at each vertex of a convex $n$-gon. If they are allowed to move together along sides or diagonals of the polygon to reach another vertex, how many different ways are ...
- 201
4
votes
2
answers
611
views
Ask for a generating function or an explicit expression of a triangle of positive integers
Preliminaries
I encountered the following triangle of positive integers:
$c_{n,k}$
$n=1$
$n=2$
$n=3$
$n=4$
$n=5$
$n=6$
$n=7$
$n=8$
$k=0$
$1$
$3$
$15$
$105$
$315$
$3465$
$45045$
$45045$
$k=1$
$5$
$...
- 1,101
4
votes
0
answers
414
views
Explicit formula for tournament sequence
I am looking for an explicit formula for a sequence. The sequence is generated as follows:
There is a tournament with $10$ teams. In the beginning, all teams have a 0-0 win-loss record. The teams are ...
- 41
4
votes
0
answers
156
views
Inequalities about tripling and doubling sumsets
Let $A$ be a set of vectors in $\mathbb Z^d$ who $\mathbb R$-span is the whole $\mathbb R^d$. Let $s_i(A)$ denote the size of $A+A+\dots A$ ($i$ times). I am interested in the following:
Question 1:...
- 30.6k
3
votes
2
answers
197
views
Limit of the Schröder numbers ratio
I have been playing around with interesting integer sequences and came across Schröder number which defines the number of lattice paths of n x n grid.
The recurrence formula to calculate these numbers ...
3
votes
2
answers
203
views
Determining the asymptotic behavior of a sequence
I've encountered the following sequences
$$
a_k=2^{k+1}\sum_{j=0}^{k-1}a_{k-1-j}a_j,\;a_0=1
$$
$$
b_k=(k+1)\sum_{j=0}^{k-1}b_{k-1-j}b_j,\;b_0=1.
$$
I would like to have an estimate of the growth of ...
- 843
3
votes
1
answer
159
views
Limit associated with two Beatty sequences that are not a Beatty pair
Suppose that $r>1$ and $s>1$ are irrational numbers, and let $a_n=\lfloor nr \rfloor$ and $b_n=\lfloor ns \rfloor$. Assume that $r$ and $s$ are numbers for which $\{a_n\}\cap\{b_n\}$ is ...
- 479
3
votes
1
answer
308
views
Tangent numbers, secant numbers and permanent of matrices
Inspired by Question 402572, I consider the permanent of matrices
$$f(n)=\mathrm{per}(A)=\mathrm{per}\left[\operatorname{sgn} \left(\sin\pi\frac{j+2k}{n+1} \right)\right]_{1\le j,k\le n},$$
where $n$ ...
- 884
3
votes
1
answer
330
views
Counting Bipartitions
Numerical evidence suggests that $p_2(n) \geq n p(n)$ for large $n$.
Here $p(n)$ is the number of partitions of $n$, and $p_2(n)$ is the number of bipartitions of $n$, i.e., ordered pairs of ...
- 133
3
votes
1
answer
298
views
Sequences with integral variances
This is a companion to my earlier question,
Sequences with integral means.
This new question is, frankly, not as interesting, but it feels necessary to complete
the thought.
Let $V(n)$ be the ...
- 151k
3
votes
1
answer
92
views
Partition of $(2^{n+1}+1)2^{2^{n-1}+n-1}-1$ into parts with binary weight equals $2^{n-1}+n$
Let $\operatorname{wt}(n)$ be A000120, i.e., number of $1$'s in binary expansion of $n$ (or the binary weight of $n$).
Let $a(n,m)$ be the sequence of numbers $k$ such that $\operatorname{wt}(k)=m$. ...
- 4,948
3
votes
1
answer
140
views
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.
$$\operatorname{exp}\left(x + \frac{\operatorname{exp}(kx) - 1}{k}\right)$$
...
- 4,948
3
votes
2
answers
285
views
Distinct distances between adjacent equal elements
Let's call a sequence $a_1, \ldots, a_n$ suitable if for any positive integer $d$ there is at most one index $i$ such that $a_i = a_{i + d}$ and all elements $a_{i + 1}, \ldots, a_{i + d - 1}$ are not ...
- 5,590
3
votes
0
answers
128
views
Fast and simple algorithm for the A329369
Let $a(n)$ be A329369 (i.e., number of permutations of $\{1,2,\cdots,m\}$ with excedance set constructed by taking $m-i$ ($0 < i < m$) if $b(i-1) = 1$ where $b(k)b(k-1)\cdots b(1)b(0)$ ($0 \...
- 4,948
3
votes
0
answers
120
views
Sequence which is related to the binary expansion of $n$ and partition numbers
Let $p(n)$ be A000041 i.e. the number of partitions of $n$ (the partition numbers).
Let
$$
\ell(n)=\left\lfloor\log_2 n\right\rfloor
$$
Let $\operatorname{wt}(n)$ be A000120 i.e. number of $1$'s in ...
- 4,948
3
votes
0
answers
69
views
Sequence that sum up to A343685
Let $a(n)$ be A343685 i.e.
$$
a(n)=2na(n-1)+\sum\limits_{j=0}^{n-1}\binom{n}{j}(n-j-1)!a(j), \\
a(0)=1
$$
Here the exponential generating function $A(x)$ satisfy
$$
A(x)=\frac{1}{1-2x+\log(1-x)}
$$
...
- 4,948
3
votes
0
answers
165
views
Closed form for $a(2^m(2^n-2^p-1))$
Let $q(n)$ be A007814, i.e., the 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
$$a(2n+1)=a(n), a(2n)=a(n)+a(n-2^{q(n)})+a(...
- 4,948
3
votes
0
answers
135
views
Permutation of a sequence, such that $y_i+y_{i+1}$ are all distinct
The sequence $x_1, x_2, ..., x_n$ of positive integers contains at least $\frac {2n}{3}+1$ distinct numbers and each of them appears at most three times. How to prove that there is a permutation $y_1, ...
- 3,153
3
votes
1
answer
91
views
Source coding lexicographic index of finite alphabet sequence with weight (partitions)
My goal is to determine the lexicographic index of an $M$-ary $n$-sequence $\mathbf{x}$ on the subset with an $M$-weight sum constraint:
$$S = \{ \mathbf{x} \in \{0, \ldots, M-1\}^n: \sum_{j=1}^n x_j =...
- 131
2
votes
2
answers
242
views
Negated Fibonacci and the floor function
Let $F_n$ be A000045 (i.e., Fibonacci numbers). Here
$$
F_n = F_{n-1} + F_{n-2}, \\
F_0 = 0, F_1 = 1, \\
F_{-n} = (-1)^{n-1}F_n
$$
I conjecture that
$$
F_{-n} = \left\lfloor\frac{n+1}{2}\right\rfloor ...
- 4,948
2
votes
2
answers
273
views
Alternating binomial-harmonic sum: evaluation request
Let $H_k=\sum_{j=1}^k\frac1j$ be the harmonic numbers.
QUESTION. Can you find an evaluation of the following sum?
$$\sum_{a=1}^b(-1)^a\binom{n}{b-a}\frac{H_{b-a}}a.$$
- 43.2k
2
votes
1
answer
217
views
Number of distinct higher dimensional integer partitions
By a distinct partition, I mean a partition into distinct parts, i.e., $10 = 5+4+1$ is one, but $10=6+2+2$ is not. The number of distinct partitions of $k$ all whose parts are at most $n$ is given by ...
- 281
2
votes
2
answers
317
views
sum of odious numbers to the power of k
In number theory, an odious number is a positive integer that has an odd number of $1$s in its binary expansion.
The first odious numbers are:
$1, 2, 4, 7, 8, 11, 13, 14, 16, 19, 21, 22, 25, 26, 28, ...
- 29
2
votes
1
answer
740
views
Power tower made of $2$s and $3$s: too high, too soon?
A power tower of a number $x$ is typified by
$$ x^{x^{x^{x^{x^{x^{x^{x^{x^x}}}}}}}}.$$
Here, however, we take the liberty of referring to the set $T$ of "$\{2,3\}$-power towers"; i.e., numbers
$$...
- 3,443
2
votes
3
answers
284
views
Making integer multisets graphic
Let $M=(X,f)$ be a multiset, where $X$ is the underlying set of elements and $f:X\rightarrow\mathbb{N}$ is the multiplicity function. For every $k\in\mathbb{N}$ put $k\cdot M:=(X,k\cdot f)$. It is ...
- 747
2
votes
1
answer
216
views
Simplification of the closed form for the A329369
Let $s(n,k)$ be a (signed) Stirling number of the first kind.
Let ${n \brace k}$ be a Stirling number of the second kind.
Let
$$
f(n,m,i) = (-1)^{m-i+1}\sum\limits_{j=i}^{m+1}j^n s(j,i) {m+1 \brace ...
- 4,948
2
votes
1
answer
153
views
Bounds for the sequence $a(n)=a(n-1)+a(\lfloor n-n^A \rfloor)$
Related to the question about a(n)=a(n-1)+a(floor(n/2))
Let $A$ be real constant $ 0 < A < 1$.
Define the sequence $a(n)$ by $a(1)=1, a(n)=a(n-1)+a(\lfloor n-n^A \rfloor)$
(if you prefer take $a'...
- 25.4k
2
votes
1
answer
146
views
On gaps in a sequence of integers
Given a fixed $p \in \{3,4,5,\ldots\}$, we define the strictly increasing sequence $\{a_k\}_{k\in \mathbb N}$ as follows. We set $a_{p,1}=1$ and for each $k>1$, we set $a_{p,k}$ to be the least ...
- 4,115
2
votes
1
answer
301
views
Number of subsets that sum to $0$
Suppose you choose $n$ distinct random numbers from a contiguous subset of cardinality $f({\beta, n})$ with at least $f({\alpha_+, n})$ positive and at least $f({\alpha_-, n})$ negative values from a ...
user76479
2
votes
1
answer
131
views
Sequence that sums up to A224071
Let $a(n)$ be A224071 (i.e., number of Schroeder paths of semilength $n$ in which there are no $(2,0)$-steps at level $1$). Here
$$
a(n) = \frac{1}{2(n+1)}\sum\limits_{k=0}^{n}(k+1)((-1)^{\left\...
- 4,948
2
votes
2
answers
178
views
Fibonacci-like sequence
Fix three integers $a, b, c$ and consider a sequence of integers $a_{i,j}$ defined, for $i \ge 0, j \ge 0$, recursively as follows:
$a_{i,0}=1$ for every $i$, $a_{0,j}=a+bj+cj^2$ and, for $i \ge 1, j \...
- 331
2
votes
1
answer
120
views
Recursion for the Chebyshev transform of $m^n$
Let
$$
R(n, q, m) = R(n-1, q+1, m) + \sum\limits_{j=0}^{q} (-1)^{q-j}R(n-1, j, m), \\
R(0, q, m) = (m-1)^q
$$
I conjecture that $R(n, 0, m)$ is a Chebyshev transform of $m^n$.
Examples of Chebyshev ...
- 4,948
2
votes
1
answer
177
views
An upper bound on coefficients of some integer sequences
Given $\lambda>0$ let $B=B(\lambda)$ be the smallest integer
such that there exist infinite integer sequences
having values in $\lbrace 1,2,\ldots,B-1,B\rbrace$ and satisfying
the following ...
- 17.9k
2
votes
1
answer
196
views
Guess (or upper bound) the general formula for a double sequence
Let $t,s \geq 0$ be integers. We have the following recursive formula:
$$f(t+1,s) = f(t,s) + f(t,s-1) + \sum_{0\leq a,b,c \leq h(t):\\a+b+c = s-1}f(t,a)f(t,b)f(t,c),$$ where
$$h(t) = \frac{1}{2}3^t -\...
- 515
2
votes
1
answer
238
views
"flavored" equivalence classes of permutations
We say two permutations $\pi_1$ and $\pi_2$ in the symmetric group $\mathfrak{S}_n$ are $k$-equivalent, denoted
$\pi_1 \sim_k \pi_2$, if one can be
determined from the other after a finite number of ...
- 43.2k