# Questions tagged [integer-sequences]

For questions about sequences of integers. References are often made to the online resource oeis.org.

319
questions

-2
votes

1
answer

136
views

### Two-variable continuous function which results in an integer if and only if arguments are integer

I am looking for functions $f(x,y)$, real arguments, continuous,
with the following properties:
$f(m,n) = r$, where $r$ is integer $> 0$ if and only if $m,n$ are integers $> 0$.
$f(m,n) \le f(...

0
votes

0
answers

49
views

### 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? ...

7
votes

0
answers

70
views

### 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 ...

3
votes

1
answer

107
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.
$$\operatorname{exp}\left(x + \frac{\operatorname{exp}(kx) - 1}{k}\right)$$
...

2
votes

0
answers

94
views

### 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
votes

1
answer

159
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,...

2
votes

0
answers

63
views

### Integer coefficients such that $T(n,k)=R(n,k)-R(n,k-1)$

Let $a(n)$ be A000085, i.e., the number of self-inverse permutations on $n$ letters, also known as involutions; number of standard Young tableaux with $n$ cells. Here
$$a(n) = a(n-1) + (n-1)a(n-2), a(...

1
vote

0
answers

49
views

### 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
vote

0
answers

130
views

### 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, \...

0
votes

0
answers

66
views

### Permutation that produces permutations

Let $f(n)$ be A000045, i.e., Fibonacci numbers: $f(n)=f(n-1)+f(n-2)$ for $n>1$ with $f(0)=0$ and $f(1)=1$.
Let $g(n)$ be A072649, i.e., $n$ occurs $f(n)$ times. The sequence begins with
$$1, 2, 3, ...

2
votes

0
answers

72
views

### Uniqueness of the permutation

Let $f(n)$ be A000045(n), i.e., Fibonacci numbers: $f(n)=f(n-1)+f(n-2)$ for $n>1$ with $f(0)=0$ and $f(1)=1$.
Let $g(n)$ be A072649, i.e., $n$ occurs $f(n)$ times. The sequence begins with
$$1, 2, ...

0
votes

0
answers

29
views

### Partitioning vectors from Z^k into bundles preserving their additive properties

Let $B_1, B_2, \dots, B_m$ be disjoint subsets of $\mathbb{Z}^k$ and $B$ denote their union.
Also suppose that $k$ upper bounds the $\ell^\infty$-norm of every vector in $B$.
A set $V \subseteq B$ of ...

1
vote

0
answers

119
views

### Ask for a proof of an inequality involving the Bernoulli numbers

Let $B_k$ be the Bernoulli numbers and let
\begin{equation}
T_k=\frac{2^{2k}}{(2k)!}|B_{2k}|, \quad k\ge1.
\end{equation}
Prove the inequality
\begin{equation*}
\frac{\frac{1}{k+2}\sum_{j=0}^{k+1}\...

1
vote

0
answers

101
views

### Existence of binary permutations with a given property

Let
$$\ell(n)=\left\lfloor\log_2 n\right\rfloor$$
Let
$$f(n)=n-2^{\ell(n)}$$
Let $a(n)$ be a permutation of the nonnegative integers such that $a(0)=0$, $a(n)=n$ if $n$ is a power of $2$ and ...

2
votes

1
answer

124
views

### Permutation and its binary analog

Let $f(n)$ be A000045(n), i.e., Fibonacci numbers: $f(n)=f(n-1)+f(n-2)$ for $n>1$ with $f(0)=0$ and $f(1)=1$.
Let $g(n)$ be A072649, i.e., $n$ occurs $f(n)$ times. The sequence begins with
$$1, 2, ...

1
vote

0
answers

71
views

### Infiniteness of the pairs of sequences with a given conditions

Let
$$\varphi=\frac{1+\sqrt{5}}{2}$$
Let
$$a_1(n)=\left\lfloor n\varphi \right\rfloor, a_2(n)=n+a_1(n)$$
Let $\operatorname{tr}(n)$ be A007814, i.e., the number of trailing zeros in the binary ...

0
votes

0
answers

42
views

### Stolarsky array and Stolarsky representation

Let $T(n,k)$ be A035506, i.e., Stolarsky array read by antidiagonals. Here we consider that $T(n,k)=0$ for $n<1, k<1$.
Let $a(n)$ be A200714, i.e., Stolarsky representation interpreted as binary ...

1
vote

1
answer

98
views

### Coefficients of number of the same terms which are arising from iterations based on binary expansion of $n$

Let
$$\ell(n)=\left\lfloor\log_2 n\right\rfloor$$
Let
$$T(n,k)=\left\lfloor\frac{n}{2^k}\right\rfloor\operatorname{mod}2$$
Here $T(n,k)$ is the $(k+1)$-th bit from the right side in the binary ...

4
votes

0
answers

375
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 ...

2
votes

2
answers

137
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 \...

1
vote

0
answers

94
views

### Subsequence such that $c(a(n))=2^n$

Let $a(n)$ be A060831, i.e., $\sum\limits_{k=1}^{n}\operatorname{number of odd divisors of} k$.
Let
$$\ell(n)=\left\lfloor\log_2 n\right\rfloor$$
Let
$$b(n,k)=2b(n,k-1)-2^{k-1}, b(n,0)=n$$
Let $c(n)$ ...

1
vote

0
answers

57
views

### Is this factorization problem in EXP?

Factorization is not known to have a polynomial time algorithm. Traditionally the input length is number of bits in representation of the integer to be factored.
However now consider integers of form $...

1
vote

1
answer

96
views

### Given a real $x>1$, construct an aperiodic substitution sequence whose complexity functions grow like $xn$

The Fibonacci word is a binary sequence defined as follows.
We use a substitution rule $0\to 01$, $1\to 0$. Then, starting with the binary string $0$, apply the substitution rules successively. So we ...

2
votes

0
answers

153
views

### Closed form for the A347205

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 $\operatorname{wt}(n)$ be A000120, i.e., number of $1$'s in binary ...

2
votes

0
answers

110
views

### Closed form for the sum of the integer coefficients

Let $a(n)$ be A002720, i.e., number of partial permutations of an $n$-set; number of $n \times n$ binary matrices with at most one $1$ in each row and column.
$$a(n)=\sum\limits_{k=0}^{n} k!\binom{n}{...

2
votes

0
answers

168
views

### Chess pieces metrics in higher dimensions

A couple of days ago, I was thinking about applying the knight (the well-known piece of chess) metric to any cubic lattice $\mathbb{N}^k$, $k \in \mathbb{N}-\{0,1\}$.
I suddenly realized that, from $k ...

2
votes

0
answers

189
views

### What do we know about Lucky numbers?

I'm really fascinated by lucky numbers (Wikipedia; OEIS A000959) and their prime-like characteristics. They have their very own Goldbach, Legendre, Lemoine and twin conjectures. I was wondering ...

0
votes

0
answers

90
views

### Closed form for the number of steps required to get $n$ balls in the last box

Let $\operatorname{wt}(n)$ be A000120, i.e., number of $1$'s in binary expansion of $n$ (or the binary weight of $n$).
Then we have an integer sequence given by
$$a(n)=n(n+1)-\sum\limits_{k=0}^{n}\...

1
vote

1
answer

100
views

### Cardinality of $\{ n_i + i^k: i \in \mathbb{N} \} \cap [1,T]$ where $\{n_i \}$ is all natural numbers in some order

Let $n_1, n_2, ...$ be a sequence of natural numbers such that $\{n_i: i \in \mathbb{N}\}$ as a set is all of natural numbers. Let $k$ be a positive integer. Is is possible to obtain a lower bound of ...

3
votes

0
answers

247
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:
$c_{n,k}$
$n=1$
$n=2$
$n=3$
$n=4$
$n=5$
$n=6$
$n=7$
$n=8$
$k=0$
$1$
$3$
$15$
$105$
$315$
$3465$
$45045$
$45045$
$k=1$
$5$
$...

0
votes

0
answers

107
views

### Two different ways to compute the same sequence (A329369)

Let $p(n,k,m)$ be the $k$-th element of the $n$-th permutation of length $m$ where permutations sorted in lexicographic order. Here $p(n,k,m)=0$ for $n>m!$.
Let
$$f(n,k,m)=[p(n,k,m)> k]$$
and ...

0
votes

1
answer

88
views

### Non-Wieferich primes with Euler quotient modulo $p$ two and alternating harmonic numbers

Let $b(n)$ denote the Euler quotient modulo $n$.
In OEIS we have A128465 Numbers k such that k divides the numerator of alternating Harmonic number H'((k+1)/2)
For $n>1$ we have $b(A128465(n))=2$.
...

0
votes

0
answers

86
views

### What will be the set of non-Wieferich numbers if the set of non-Wieferich primes is finite?

Integer $n$ is Wieferich number if $2^{\phi(n)}-1 \equiv 0 \pmod {n^2}$.
Wieferich prime is Wieferich number with $n$ prime.
It is an open problem if there are infinitely many Wieferich primes
and ...

3
votes

0
answers

103
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(...

2
votes

1
answer

90
views

### Asymptotic analysis of a peculiar sum of squares sequence

Let $a,b$ be two positive integers. Let the sequence $\{s_n\}_n$ be the set of all possible sums of squares $a^2+b^2$, such that they are in ascending order
\begin{align*}
& n=1 & s_1=1^2+1^2=...

0
votes

1
answer

182
views

### Are there infinitely long arithmetic progressions in every increasing sequence of positive integers with bounded gaps between consecutive terms?

Suppose the largest gap is D>1 and at least two of the gaps 1,2,...,D appear infinitely many times. I think the answer is NO. But I find it difficult to formulate a necessary and sufficient ...

0
votes

0
answers

100
views

### 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
votes

1
answer

89
views

### 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 ...

1
vote

0
answers

64
views

### 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 $2$ ...

5
votes

0
answers

189
views

### How to solve the recursive formula $$A(n,k)=A(n-1,k)+A(n,k-1)+A(n-1,k-1)$$

Is there any known solution for the recursive formula
$$A(n,k)=A(n-1,k)+A(n,k-1)+A(n-1,k-1)$$
for given initial values A(0,0), A(1,0) and A(0,1)?
Does this formula have any geometric or combinatorial ...

1
vote

0
answers

65
views

### Conjecture on numbers $k$ having only one partition into parts with same binary weight as a binary weight of $k$

Let $\operatorname{tr}(n)$ be A007814, number of trailing zeros in the binary representation of $n$.
Also, let $\operatorname{ntr}(n)$ be A086784, number of non-trailing zeros in the binary ...

3
votes

1
answer

85
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 $\operatorname{wt}(k)=m$. ...

0
votes

1
answer

116
views

### Weirdness in the sequence "the number of divisors for a weird number"

I thought it would be fun to give my froshling students a short programming assignment to characterize numbers as: deficient, abundant, perfect, and prime. Then I got a little carried away and started ...

1
vote

0
answers

61
views

### Counting pieces when an object is cut n ways

I was reading a passage from an old essay by Martin Gardner on the calculus of finite differences, and it seems to me that there is more that can and should be said about seemingly anomalous values of ...

2
votes

1
answer

125
views

### Is there a way to find all number series whose formulae of general term contain progressions?

Let $\{c_{m,n}\}_{m,n\in\mathbb{N}}$ be known complex numbers. My question is, how to find all number series $\{a_{n}\}_{n\in\mathbb{N}}$ such that
$$a_n=\sum_{m=0}^\infty c_{m,n}a_{m+n},~\forall n\...

1
vote

1
answer

434
views

### How many non-isomorphic, simple, connected graphs with 6 vertices are there? [closed]

A graph is called simple if there are no loops and there are no multiple edges. Is it possible to compute the number of non-isomorphic, simple, connected graphs with 6 vertices? If the number is known,...

0
votes

0
answers

129
views

### What can we say about the following number sequence?

$\{b_n\}_{n\geq0}$ is a number sequence satisfying the following condition:
\begin{equation}
b_{m}=\sum_{r=0}^m\sum_{h=0}^r\left(\frac{m!}{(m-r)!(r-h)!h!}\right)^2b_{m+h-r}b_{r},~\forall m\in\...

7
votes

2
answers

743
views

### Distance among integer set

Given an integer set, if the distances between integers in the set are still in the set, what mathematical term should be used to describe that nature? Or what nature does the set have?
For example, $...

1
vote

1
answer

146
views

### On the sequence $a(n)=\gcd(2^n-1,\phi(2^n-1))$

For natural $n$, define the sequence
$$
a(n)=\gcd(2^n-1,\phi(2^n-1))
$$
It doesn't appear to be in OEIS and starts
$1,1,1,1,9,1,1,1,3,1,9,1,3,1,1,1,27,1,75,49$
Q1 Can we unconditionally prove $a(n)=1$...

3
votes

1
answer

153
views

### Are there infinitely many nonzero Euler quotients $a(n)=\frac{2^{\phi(n)}-1}{n} \bmod n$?

This might be related to an open problem.
For odd natural $n$ define the Euler quotient:
$$ a(n)=\frac{(2^{\phi(n)}-1) \bmod n^2}{n}=\frac{2^{\phi(n)}-1}{n} \bmod n$$
Q1 Are there infinitely many $n$ ...