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Questions tagged [integer-sequences]

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

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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(...
Jada's user avatar
  • 3
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? ...
TheSimpliFire's user avatar
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 ...
Max Alekseyev's user avatar
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. $$\operatorname{exp}\left(x + \frac{\operatorname{exp}(kx) - 1}{k}\right)$$ ...
Notamathematician's user avatar
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 ...
Notamathematician's user avatar
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,...
Notamathematician's user avatar
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(...
Notamathematician's user avatar
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, \...
Notamathematician's user avatar
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, \...
Notamathematician's user avatar
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, ...
Notamathematician's user avatar
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, ...
Notamathematician's user avatar
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 ...
Michal's user avatar
  • 33
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}\...
qifeng618's user avatar
  • 706
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 ...
Notamathematician's user avatar
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, ...
Notamathematician's user avatar
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 ...
Notamathematician's user avatar
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 ...
Notamathematician's user avatar
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 ...
Notamathematician's user avatar
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 ...
Jackson's user avatar
  • 41
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 \...
Cob's user avatar
  • 131
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 $c(n)$ ...
Notamathematician's user avatar
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 $...
Turbo's user avatar
  • 13.2k
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 ...
Darren Ong's user avatar
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 ...
Notamathematician's user avatar
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}{...
Notamathematician's user avatar
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 ...
Marco Ripà's user avatar
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 ...
Happydugongo's user avatar
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}\...
Notamathematician's user avatar
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 ...
Johnny T.'s user avatar
  • 3,373
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: $c_{n,k}$ $n=1$ $n=2$ $n=3$ $n=4$ $n=5$ $n=6$ $n=7$ $n=8$ $k=0$ $1$ $3$ $15$ $105$ $315$ $3465$ $45045$ $45045$ $k=1$ $5$ $...
qifeng618's user avatar
  • 706
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 ...
Notamathematician's user avatar
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 $b(A128465(n))=2$. ...
joro's user avatar
  • 23.8k
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 ...
joro's user avatar
  • 23.8k
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(...
Notamathematician's user avatar
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=...
TheVal's user avatar
  • 151
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 ...
Kai Wang's user avatar
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^{...
Notamathematician's user avatar
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 ...
Notamathematician's user avatar
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 $2$ ...
Notamathematician's user avatar
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 ...
Nan's user avatar
  • 81
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 ...
Notamathematician's user avatar
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 $\operatorname{wt}(k)=m$. ...
Notamathematician's user avatar
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 ...
Prester John's user avatar
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 ...
James Propp's user avatar
  • 18.8k
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\...
Ren Guan's user avatar
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,...
John Depp's user avatar
  • 147
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\...
Ren Guan's user avatar
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, $...
hui cj's user avatar
  • 79
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 $a(n)=1$...
joro's user avatar
  • 23.8k
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 $n$ ...
joro's user avatar
  • 23.8k

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