All Questions
Tagged with integer-sequences co.combinatorics
160 questions
35
votes
8
answers
3k
views
Examples of integer sequences coincidences
For the time being, the OEIS website contains almost $300000$ sequences. Each of these sequences is the mark of a specific mathematical concept. Sometimes two (or more) distinct concepts have the ...
33
votes
0
answers
2k
views
The easily bored sequence
If we want to compare the repetitiveness of two finite words, it looks reasonable, first of all, to consider more repetitive the word repeating more times one of its factors, and secondarily to ...
- 4,459
27
votes
1
answer
891
views
Why do the adjoint representations of three exceptional groups have the same first eight moments?
For a representation of a compact Lie group, the $n$th moment of the trace of that representation against the Haar measure is the dimension of the invariant subspace of the $n$th tensor power. The ...
- 149k
26
votes
1
answer
7k
views
Elegant recursion for A301897
Let $a(n)$ be A301897, i.e., number of permutations $b$ of length $n$ that satisfy the Diaconis-Graham inequality $I_n(b) + EX_n(b) \leqslant D_n(b)$ with equality. Here
$$a(n)=\frac{1}{n+1}\binom{2n}{...
- 4,948
26
votes
3
answers
907
views
What is the smallest size of a shape in which all fixed $n$-polyominos can fit?
Let $n$ be an integer and consider all fixed $n$-polyominos, i.e., without rotation or reflection. I am interested in finding a shape in which all polyominos can embed. (It is OK if multiple ...
- 431
23
votes
3
answers
2k
views
Zeroes of the random Fibonacci sequence
Let $X_n$ be the "random Fibonacci sequence," defined as follows:
$X_0 = 0, X_1 = 1$;
$X_n = \pm X_{n-1} \pm X_{n-2}$, where the signs are chosen by independent 50/50 coinflips.
It is known ...
- 19.2k
23
votes
5
answers
1k
views
Sequences with integral means
Let $S(n)$ be the sequence whose first element is $n$, and from then onward,
the next element is the smallest natural number ${\ge}1$ that ensures that the
mean of all the numbers in the sequence is ...
- 151k
20
votes
2
answers
1k
views
A possibly surprising appearance of $\sqrt{2}.$
Define $A=(a_n)$ and $B=(b_n)$ as follows: $a_0=1$, $a_1=2$, $b_0=3$, $b_1=4$, and $$a_n=a_1b_{n-1}-a_0b_{n-2} + 2n$$ for $n \geq 2$, where $A$ and $B$ are increasing and every positive integer occurs ...
- 3,443
19
votes
2
answers
2k
views
A finite alternating sum
We have stumbled upon the following finite alternating sum, which we have trouble analyzing. The sum is:
$$
S_n = \sum_{j=0}^n \frac{ (-1)^j e^{-j} }{j!} (n-j)^j
$$
We have observed numerically that ...
- 193
19
votes
2
answers
581
views
Sequences with 3 letters
For a positive integer $n$ I would like to construct long sequences consisting of 0, 1 and 2's such that for any two subsequences consisting of $n$ consecutive elements the number of 0's , 1's or 2'...
- 2,286
18
votes
2
answers
992
views
A conjecture harmonic numbers
I will outlay a few observations applying to the harmonic numbers that may be interesting to prove (if it hasn't already been proven).
From the Online Encyclopedia of Positive Integers we have:
$a(n)$ ...
- 181
18
votes
1
answer
607
views
Order of Conway's "look and say" recurrence
Let $L_n$ be the length of the $n$th term of Conway's "look and say"
sequence (https://oeis.org/A005341). The generating function $F(x)=
\sum_{n\geq 0}L_nx^n$ is a rational function, say $P(x)/Q(x)$ ...
- 50.8k
16
votes
2
answers
1k
views
are these polynomials or rationals functions?
Let $x$ be a variable. Define the following family of sequences (reminiscent of Lucas polynomials) according to the rule: $P_0(x):=0, P_1(x):=1$ and for $n\geq2$ by
$$P_n(x)=xP_{n-1}(x)-P_{n-2}(x).$$
...
- 43.2k
15
votes
1
answer
1k
views
Arithmetic progressions in stopping time of Collatz sequences
Inspired by the question here, we did a few more simulations of numbers of some specific forms and noticed a pattern.
We consider the original $3n+1$ transform where we divide by $2$ if it's even and ...
- 259
15
votes
0
answers
487
views
Word complexity of primes mod 4
For an infinite binary word $w$, the word complexity $f_w(n)$ is defined as the number of different subwords of length $n$. The asymptotic behavior of this function is an important parameter of the ...
- 17.1k
14
votes
4
answers
2k
views
Integrality of a sequence formed by sums
Consider the following sequence defined as a sum
$$a_n=\sum_{k=0}^{n-1}\frac{3^{3n-3k-1}\,(7k+8)\,(3k+1)!}{2^{2n-2k}\,k!\,(2k+3)!}.$$
QUESTION. For $n\geq1$, is the sequence of rational numbers $a_n$ ...
- 43.2k
14
votes
1
answer
697
views
Are the asymptotics of A003238 known?
Sequence A003238 of the OEIS counts ``rooted trees with $n$ vertices in which vertices at the same level have the same degree.'' The sequence, $a$, begins
1, 1, 2, 3, 5, 6, 10, 11, 16, ...
and it is ...
- 1,305
14
votes
1
answer
4k
views
Put as many points as possible in an equilateral triangle of side 1 with their minimal distance greater than 1/n
It is known by the pigeon-hole principle that:
If we select $5$ points within an equilateral triangle with side $1$, there must be at least two whose distance apart is less than or equal to $1/2$.
...
- 241
14
votes
1
answer
835
views
Special configurations on a circle from a homological algebra problem
Here is the short version of the combinatorial problem:
Given a positive integer $n \geq 2$. Draw a circle with $2n$ points indexed by the numbers from $\mathbb{Z}/ 2n \mathbb{Z}$. We colour the ...
- 26.5k
13
votes
2
answers
2k
views
Positive integers written as $\binom{w}2+\binom{x}4+\binom{y}6+\binom{z}8$ with $w,x,y,z\in\{2,3,\ldots\}$
Let $\mathbb N=\{0,1,2,\ldots\}$. Recall that the triangular numbers are those natural numbers
$$T_x=\frac {x(x+1)}2\quad \text{with}\ x\in\mathbb N.$$
As $T_x=\binom{x+1}2$, Gauss' triangular number ...
- 15.6k
12
votes
1
answer
427
views
Subwords of the infinite Fibonacci word
Let $W = 01001010010010 \ldots$ be the infinite Fibonacci word, A003849 in the OEIS. Let $B(m)$ be the set of $m+1$ subwords of $W$ that have length $m$, and for each such subword $u$, let $p(u)$ be ...
- 479
9
votes
2
answers
546
views
Can you tie up these Laurent sequences?
Fix an integer $k\geq3$. Define the two families of sequences $\{x_n\}$ and $\{y_n\}$ according to the rules:
$$x_n=\frac{x_{n-1}^2+x_{n-2}^2+\cdots+x_{n-k+1}^2}{x_{n-k}} \qquad n\geq k$$
and
$$y_n=\...
- 43.2k
9
votes
0
answers
304
views
Symmetric function transition matrix and a non-conjecture by Clifford and Stanley
Consider the transition matrix $R = \left(R_{\lambda,\mu}\right)$, defined by
$$
p_\lambda = \sum_{\mu} R_{\lambda\mu}m_\mu ,
$$
between the power-sum and the monomial basis of the ring of symmetric ...
- 15.8k
9
votes
0
answers
398
views
When do almost all these invariants of tensors vanish?
Let $A,B,C,D$ be $n$-dimensional vector spaces over a field $k$.
There is a natural homomorphism from the $mn^m$th tensor power $A^{\otimes (m n^m)} $ of $A$ to $k$ given by the determinant map $A^{\...
- 149k
8
votes
4
answers
520
views
"Upside-down unimodal" sequences in combinatorics
Recall a sequence $a_0,\ldots,a_n$ of positive integers is unimodal if $a_0 \leq \cdots \leq a_m \geq \cdots \geq a_n$ for some $0 \leq m \leq n$. Unimodal integer sequences are abundant in ...
- 24.2k
8
votes
1
answer
364
views
Is the permanent of the matrix $[(\frac{i+j}{2n+1})]_{0\le i,j\le n}$ always positive?
Recall that the permanent of an $n\times n$ matrix $A=[a_{i,j}]_{1\le i,j\le n}$ is defined by
$$\operatorname{per}A=\sum_{\sigma\in S_n}\prod_{i=1}^n a_{i,\sigma(i)}.$$
In 2004, R. Chapman [Acta ...
- 15.6k
8
votes
4
answers
755
views
Upper bound on length of addition chain
An addition chain for $n$ is a finite sequence of integers starting at 1 and ending at $n$, such that each element is a sum of two previous elements. A short addition chain for $n$ can be used, for ...
- 3,559
8
votes
4
answers
1k
views
A Pascal's-triangle -like random process
I was exploring Pascal's triangle on a cylinder when I encountered this puzzle-like problem.
It is surely elementary, but perhaps weekend-entertaining.
Start with a permutation of $(1,2,3, \ldots, n)$...
- 151k
8
votes
0
answers
237
views
Sequences for which $\prod (1-z^n)^{a(n)}$ is a polynomial
This is mostly a reference request.
I'm working with complex coefficients, although all I have in mind have integer coefficients.
Let $a=(a(n))_{n\ge 1}$ be a sequence, say of integers (I have non-...
- 63.9k
7
votes
2
answers
964
views
Maximal number of edges and triangular cells for n points in a triangular lattice
Consider a subset of $n$ points in an equilateral triangular lattice. Draw all the edges between nearest-neighbor points.
What is the maximum, over all such subsets, of the number of edges? This ...
- 567
7
votes
2
answers
428
views
Limit associated with complementary sequences
Define $A=(a_n)$ and $B=(b_n)$ as follows: $a_0=1$, $a_1=2$, $b_0=3$, $b_1=4$, and $$a_n=a_0b_{n-1}+a_1b_{n-2}$$ for $n \geq 2$, where $A$ and $B$ are increasing and every positive integer occurs ...
- 3,443
7
votes
1
answer
527
views
Suitable closed form for the A079501
Let $a(n)$ be A079501 (i.e., number of compositions of the integer $n$ with strictly smallest part in the first position).
The sequence begins with
$$
1, 1, 2, 2, 4, 5, 8, 12, 19, 28, 45, 70, 110, ...
- 4,948
7
votes
1
answer
386
views
Closed form expression for a recursion relation with binomial coefficients
I am interested in the following sequence: $$ T_n = \sum\limits^{n-1}_{k=0} \begin{pmatrix} n \\ k \end{pmatrix} T_{k}, \ \ \ \ T_0 = C \in \mathbb{N} $$
I would like to express it as a function of n, ...
- 71
7
votes
1
answer
455
views
More asymptotics for trees
This is a follow up to my recent question on the asymptotics of A003238. Lucia gave a fine answer to that question, but as I hinted the 'real' problem I have in mind is slightly different, and I've ...
- 1,305
7
votes
1
answer
428
views
Counting hyperplane arrangements up to combinatorial equivalence, simple examples and history
Two arrangements of (affine) hyperplanes in $d$-dimensional Euclidean space are combinatorially isomorphic (or combinatorially equivalent) if they have isomorphic posets of faces.
Counting the ...
7
votes
0
answers
147
views
Factor-counting sequence
Define a non-negative integer sequence $\{\mathcal{F}_n\}$ as follows: start with 1 and, at each step, insert the number of entries already present in the sequence which are factors of the last one.
...
- 4,459
7
votes
0
answers
184
views
Some conjectural congruences involving Domb numbers
The Domb numbers are given by
$$D_n=\sum_{k=0}^n\binom{n}{k}^2\binom{2k}k\binom{2(n-k)}{n-k}\ \ \ (n=0,1,2,\ldots).$$
Such numbers have combinatorial interpretation, see, e.g., http://oeis.org/A002895....
- 15.6k
6
votes
5
answers
546
views
Bounds for $a(n)=a(n-1)+a(\lfloor n/2 \rfloor)$
This is related to problem in graph theory.
OEIS defines A033485 as
$a(1)=1$ and $a(n)=a(n-1)+a(\lfloor n/2 \rfloor)$.
Q1 what are upper bounds and asymptotics for $a(n)$, can we get $\exp(o(n))$?
...
- 25.4k
6
votes
2
answers
741
views
Shifting an irrational binary sequence
Let $\newcommand{\tn}{\{0,1\}^\mathbb{N}}\tn$ be the collection of all infinite binary sequences. For $s\in\tn$ and $k\in\mathbb{N}$ let the left-shift of $s$ by $k$ positions, $\ell_k(s)\in \tn$, be ...
- 48.9k
6
votes
1
answer
367
views
On A057985 and A287066
Let $a(n)$ be A057985 (i.e., start with $0$ and repeatedly substitute: $0 \to 01$, $1 \to 12$, $2 \to 0$).
Let $b(n)$ be A287066 (i.e., start with $1$ and repeatedly substitute: $0 \to 01$, $1 \to 12$...
- 4,948
6
votes
1
answer
282
views
Integer sequences with a periodic pattern
Let $A$ and $B$ be two different integers. Let $S$ be a finite integer sequence with exactly $n_A$ $A$s and $n_B$ $B$s. By repeating $S$ infinitely many times we obtain an infinite integer sequence $P$...
- 93
6
votes
1
answer
393
views
Test for pair of odd primes $(p, 2p^2-1)$
Let $a(n)$ be A106483 (i.e., primes $p$ such that $2p^2-1$ is also prime).
Let $b(n)$ be an integer sequence such that $b(n) = B$ after the whole transformation where we start with $A = n$, $B = 1$, $...
- 4,948
6
votes
1
answer
173
views
$\omega$-de-Bruijn sequences
Let $\omega$ denote the set of non-negative integers. For which integers $n>1$ is there a sequence $b_n: \omega\to\omega$ with the following property?
Whenever $v\in\omega^n$ there is a unique $...
- 48.9k
6
votes
1
answer
281
views
Is this Laurent phenomenon explained by invariance/periodicity?
In Chapter 4 (page 23, subsection "Somos sequence update") of his Tracking the Automatic
Ant, David Gale
discusses three families of recursively defined sequences of numbers, all
due to Dana ...
- 33.8k
6
votes
1
answer
268
views
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,...
- 4,948
6
votes
1
answer
224
views
Sequence A76132 eventually periodic modulo $2,3$ and $5$
Sequence A76132 starting as $1,1,2,4,10,36,218,\ldots$ of the OEIS is recursively defined by $a(1)=1$
and $a(n)=\sum_{k=1}^{n-1}a(n-k)^k$ for $n\geq 2$.
It is eventually periodic of period 1,1 and 34 ...
- 17.9k
6
votes
0
answers
171
views
An inequality involving integer partitions
For integers $n\ge k\ge0$, let $p(n,k)$ denote the number of ways to write $n$ as a sum of $k$ positive integers (repetition allowed). For example, $p(6,3)=3$ since
$$6=1+1+4=1+2+3=2+2+2.$$
QUESTION. ...
- 15.6k
6
votes
0
answers
245
views
Searching for a proof of the pattern and identification of integer coefficients for the A329369
Please see the update given below. Everything you need to know from the old version of the question are the functions $a(n), \ell(n), s(n), t(n), r(n)$.
Let $a(n)$ be A329369 (i.e, number of ...
- 4,948
6
votes
0
answers
669
views
Number of Configurations in the optimal Hanoi tower
There is a unique strategy how to move $n$ disks from the first rod to the second optimally and it takes $2^n-1$ steps, solution is obtained by simple recursion. I am interested into the following ...
- 399
5
votes
2
answers
237
views
Are the Gessel sequence integers composite for all $n\ge 3$?
The Gessel sequence is known for Ira Gessel's Lattice Path Conjecture of $2001$, which has been proved by Kauers, Koutschan and Zeilberger in $2009$ with the aid of a computer. Later, other proofs ...
- 12.1k