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For questions related to the On-Line Encyclopedia of Integer Sequences.

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Sequence $\{a_n\}$ where $a_n$ is the smallest proper multiple of $a_{n-1}$ containing $a_{n-2}$ in its digits

Recently, I've been interested in sequences $\{a_n\}$ where $a_n$ is the smallest proper multiple (multiple of a number larger than itself) of $a_{n-1}$ containing $a_{n-2}$ in its digits. A number $Y$...
Bryle Morga's user avatar
4 votes
0 answers
84 views

Is it known whether or not OEIS sequence A091308 is finite?

OEIS sequence A038395 is defined as the concatenation of the first $n$ odd numbers in reverse order: $1,$ $31,$ $531,$ $7531,$ $97531,$ $1197531,$ etc., and A091308 is the sequence of primes in ...
Ted Hopp's user avatar
  • 635
6 votes
0 answers
96 views

Factorizations not sharing digits with original number

The sequence A371862 is "Positive integers that can be written as the product of two or more other integers, none of which uses any of the digits in the number itself." In the extended ...
Ed Pegg's user avatar
  • 21.2k
3 votes
1 answer
133 views

Orientable necklaces and complements

I'm trying to understand the OEIS sequence A059078: Number of orientable necklaces with 2n beads and two colors which when turned over produce their own color complement. 0, 0, 0, 1, 2, 6, 12, 27, 54,...
Throckmorton's user avatar
-3 votes
1 answer
182 views

What is the number of integers divisible by either 2, 3, or 5 from the integers 1 to $n+1$? [closed]

I am interested in integer sequence A254828 (https://oeis.org/A254828), but from the link it seems to have a recursive formula $a(n) = a(n-1) + a(n-30) - a(n-31)$. As joriki said, they are the number ...
licheng's user avatar
  • 2,306
4 votes
4 answers
467 views

Dodecahedral number visualization

The dodecahedral numbers, 0, 1, 20, 84, 220, 455, 816, 1330, 2024, 2925, 4060, ... numbers of the form ${3 n \choose 3}$ (A006566). Does anyone have a good visualization of these? In particular, I'd ...
Ed Pegg's user avatar
  • 21.2k
2 votes
1 answer
52 views

Examples of palindromic polynomial sequences

In the OEIS you can find the coefficients of the h-polynomials for the associahedra, the Narayana polynomials A001263; for the stellahedra, A046802; and for the permutahedra, the Eulerian polynomials ...
Tom Copeland's user avatar
  • 2,426
0 votes
1 answer
45 views

Proving of an E.G.F

We wish to prove that the exponential generating function of the sequence defined by the recurrence relations $$a_0 = 1, a_1 = 1$$ $$a_n = na_{n-1} + (n-1)a_{n-2}$$ is given by $\frac{e^{-x} }{(1-x)^2}...
joannie's user avatar
  • 11
3 votes
1 answer
292 views

Understanding OEIS A139383

Because it resembles another thing that I am counting, I would like to understand the OEIS sequence A139383, "Number of $n$-level labeled rooted trees with $n$ leaves". I think I ...
Fabius Wiesner's user avatar
3 votes
1 answer
78 views

Number of Cycles from a Permutation

Let $P(n)$ be the number of cycles in a $2^n$-length permutation where the odd numbers come first and then the even numbers. For example, the permutation from $P(3)$ would be $\{1, 3, 5, 7, 2, 4, 6, 8\...
FaranAiki's user avatar
  • 297
2 votes
0 answers
47 views

Discrepancy in the number of simplices in the barycentric subdivision of an $n$-simplex

According to OEIS, the number of simplices in the barycentric subdivision of an $n$-simplex is either given by the sequence: (A002050): 1, 5, 25, 149, 1081, 9365, 94585, 1091669, 14174521, 204495125, ...
pyridoxal_trigeminus's user avatar
9 votes
3 answers
255 views

Possible new formula for OEIS A191522

The following formula (proved here): $$\sum_{k=\lfloor \frac{n+1}{2} \rfloor}^{n}{k{k-1 \choose \lfloor \frac{n+1}{2}\rfloor - 1}} = \Big\lceil \frac{n}{2} \Big\rceil{n+1 \choose \lfloor \frac{n}{2} \...
Fabius Wiesner's user avatar
0 votes
0 answers
58 views

Formula for the Wythoff array from the infinite Fibonacci word

I have a question pertaining to whether the following way of writing the Wythoff array obtained from the Fibonacci sequence / infinite Fibonacci word is known or obvious. Let $n$ be an integer. Let $F(...
Vincent Russo's user avatar
2 votes
0 answers
33 views

Floor function and hexagonal numbers

While playing around with squares, I wondered about the sum of square roots of all natural numbers between two perfect squares(both inclusive). After taking the floor value of the expression for first ...
Amrit Awasthi's user avatar
2 votes
1 answer
73 views

Explain this OEIS sequence description involving array subblocks (A227259)

Can anyone explain to me what the description of https://oeis.org/A227259 means? Number of n X 2 0,1 arrays indicating 2 X 2 subblocks of some larger (n+1) X 3 binary array having a sum of two or ...
Daniel Martin's user avatar
7 votes
2 answers
435 views

Is this a new representation of (some) Bernoulli numbers?

Let $\operatorname{B}(n)$ denote the Bernoulli numbers and $\operatorname{b}(n) = \operatorname{B}(n)/n$ with $b(0)=1$ the divided Bernoulli numbers. Also let $\sigma_{k}(n)= \sum_{d \mid n} d^k$ ...
Peter Luschny's user avatar
1 vote
0 answers
48 views

What mean $0^d$ in this sequence from OEIS

From this OEIS Sequence: $a(n+1) = (d+0^d)*10^{\lfloor{(log_{10}{a(n)+1))}}\rfloor} + (1-0^d)*\lfloor(a(n)/10)\rfloor$, where $d \equiv (a(n)+1) \,(mod \, 10)$ I want to know what that "$0^d$&...
MathLearner's user avatar
0 votes
1 answer
33 views

How to get equation from OEIS (for a zigzag repeating sequence)

While trying to find an answer to my own Stack Overflow question, I came upon this helpful math Stack Exchange where the person who answered referred to OEIS and provided an equation. I entered my ...
ESS's user avatar
  • 1
1 vote
1 answer
172 views

Normal distributions from the primes and its generation function

I am trying to understand sequence A239738 from OEIS in more detail. Basically it states: Normal distributions from the primes. The contracted sequence of integers generated from the frequencies of ...
Mikhail Gaichenkov's user avatar
2 votes
1 answer
67 views

Comparing two sequence via their exponential generating function

I am studying two sequence with their e.g.f. The first one are the Bell numbers (sequence $A000110$ on OEIS) defined as follows. $B_0=1$ and $$ B_n =\sum_{k=0}^{n-1} {n-1\choose k} B_k $$ They have a ...
Bric's user avatar
  • 133
3 votes
2 answers
163 views

Question about the name of a sequence in OEIS "Expansion of e.g.f."

I was browsing through the OEIS when I noticed the name of the sequence A052634 "Expansion of e.g.f. 1/((1-2x^2)(1-x))". I would like to know what is the expansion of e.g.f and how can I ...
CACM6's user avatar
  • 115
5 votes
0 answers
217 views

What is the growth rate of OEIS A347913?

OEIS A347913 is an extremely interesting sequence about multisets of integers. It is defined as the number of multisets one can get starting with a multiset of $n$ zeros and "splitting", ...
mathlander's user avatar
  • 4,047
13 votes
2 answers
327 views

What is the fractal dimension of the image given by a combinatorial sequence about permutation cycles?

OEIS sequence A186759 is a triangle read by rows: $T(n,k)$ is the number of permutations of $\{1,2,\dots,n\}$ having $k$ nonincreasing cycles or fixed points, where a cycle $(b_1\ b_2\ \cdots\ b_m)$ ...
Peter Kagey's user avatar
  • 5,062
-1 votes
1 answer
169 views

Solving euler problem 276

i am trying to solve Euler problem 276 But... i am completely stuck. I need to find integer triplets $a$, $b$, $c$ such that $a \leq b \leq c$ and $a+b > c$ and $\gcd\bigl(a, (\gcd(b, c)\bigr) = 1$....
user1471827's user avatar
5 votes
1 answer
513 views

Why are there lots of groups with order $2^n$?

I've noticed that, in the OEIS sequence A000001, lots of record high values are held by powers of $2.$ The records are held by only $1, 4, 8, 16, 24, 32, 48, 64, 128, 256, 512,$ and $1024.$ The only ...
mathlander's user avatar
  • 4,047
10 votes
0 answers
256 views

Positive integer solutions of $ab+1=x^2, ac+1=y^2, bc+1=z^2, x+z=2y$

A question essentially the same a this one was asked in MSE 4479792 without any background details and was deleted by the post author after I posted a minimal one sentence answer mentioning five OEIS ...
Somos's user avatar
  • 35.6k
3 votes
2 answers
134 views

Counting endofunctions by inclusion–exclusion

While leafing through the OEIS, I noticed the following conjecture (from Werner Schulte, OEIS A000312 ): For integer $n \ge 0$ $$ \sum_{k=0}^n (-1)^{n - k} { n \brack k }{n+k \brace k} = n^n . $$ ...
Peter Luschny's user avatar
0 votes
1 answer
160 views

Is there really no way to generate infinitely many primes?

Is there really no way to generate infinitely many primes? A previous answer for someone asking about the Infinite generation of primes, says: There is no exact way to generate primes continuously. ...
Malady's user avatar
  • 207
9 votes
1 answer
247 views

Counting permutations $\pi \in S_n$ such that $\pi(i) \neq i$ and $\pi(i) - k \not\equiv i \bmod n$.

Given integers $n > 2$ and $1 \leq k < n$, I'm interested in the set of permutations $$ \{\pi \in S_n : \pi(i) \neq i \text{ and } \pi(i) - k \not\equiv i \bmod n\}. $$ When $k = 1$, these are ...
Peter Kagey's user avatar
  • 5,062
7 votes
2 answers
281 views

Asymptotic for $\sum_{k=1}^n k^n$

Consider the OEIS sequence A031971, which is defined as: $$a_n=\sum\limits_{k=1}^n k^n\quad\color{gray}{(1,\,5,\,36,\,354,\,4425,\,67171,\,1200304,\,.\!.\!.\!)}\tag{1}$$ I'm interested in the ...
Vladimir Reshetnikov's user avatar
21 votes
1 answer
346 views

How many lattices does it take to cover a regular $n$-gon?

Given some positive integer $n\ge 3$, we can ask how many 2-dimensional lattices $L_1,\ldots,L_k$ are required such that their disjoint union contains all vertices of a regular $n$-gon. (We don't ...
RavenclawPrefect's user avatar
1 vote
1 answer
58 views

Question about the least prime $p$ such that $p+2n$ is also prime.

In the sequence Least prime $p$ such that $p+2n$ is also prime page (A020483) on OEIS, it says: If $a(n)$ exists, $a(n) < 2n$ What does it mean? At first it sounds like if both $p$ and $2n+p$ are ...
François Huppé's user avatar
3 votes
1 answer
72 views

Pattern in number of digits in $(10^n)!$

I was observing the OEIS sequence A061010 and its values up to $n=1000$, and it seems that the pattern is as follows. For $n$, $f(n)$ is for the form string of $(n-1)$+(pattern of repeating digits ...
Neel Shukla's user avatar
0 votes
1 answer
71 views

Recurrence relation to solve

I have one problem that can be transformed into recurrence relation shown below, I guess $a_{n} = O(n)$. But I cannot solve the analytic form of $a_{n}$, can anyone help me on this? $$(n-2)a_{n+1}-(n-...
Yuan Ji's user avatar
  • 113
0 votes
1 answer
65 views

Understanding formula for OEIS A334742: a(A033638(n)) = a(A002620(n))

How do I interpret the formula for OEIS A334742 (https://oeis.org/A334742), which is given as: $$a(A033638(n)) = a(A002620(n)) \,\,\mathrm{for}\,\, n > 1.$$ Since $$A002620(n)= \Bigl\lfloor\frac{n^...
theorist's user avatar
  • 363
3 votes
2 answers
87 views

An analogue to OEIS for graphs and ordered pairs?

Does anyone know if such a thing exists already? I was able to find a number of seemingly very comprehensive encyclopedia-like reference pages on many species of graphs, but it seemed that the use ...
Trevor's user avatar
  • 6,004
2 votes
0 answers
96 views

Lyndon words over the binary alphabet that begin with "00"

Lydon words over the binary alphabet are strings of $1$s and $0$s such that all rotations of the word are strictly lexicographically later. I've been looking at Lyndon words of length $n$, which are ...
Peter Kagey's user avatar
  • 5,062
2 votes
0 answers
376 views

Understanding first hitting time probabilities for a $2d$ random walk

The probability for a random walk that starts at the origin to return to the origin for the first time after exactly $2n$ steps has been solved on this site a couple times for the $1$d random walk. ...
user196574's user avatar
  • 1,730
1 vote
0 answers
319 views

Probability distribution of number of returns after $2n$ steps of $1d$ random walk

The expected number of returns of a simple random walk in $1d$ after $2n$ steps is a well known problem, with answer $\sum_{m=1}^n \frac{\binom{2m}{m}}{2^{2m}}$. A question that requires more ...
user196574's user avatar
  • 1,730
2 votes
1 answer
86 views

OEIS A333017, or something else entirely?

I am trying to find out how to calculate the number of different relationships there are between all the non-empty cliques that can be made of a given number 'n' of individuals. Doing this manually (...
Konchog's user avatar
  • 193
1 vote
1 answer
262 views

Regarding OEIS and integer sequences

The OEIS, a collection that was begun by Neil J. A. Sloane, is an online databse of over $360,000$ integer sequences. Now, suppose I have an integer sequence, say, $a_1, a_2, a_3, ...$ I look for ...
Firdous Mala's user avatar
  • 2,288
1 vote
1 answer
58 views

Does replacing each prime factor of an odd abundant number with the preceding prime always give another abundant number?

Given an odd abundant number $n$, if one replaces each prime factor of $n$ with the preceding prime while maintaining the same multiplicity (which gives A064989($n$)), does one always get another ...
Geoffrey Trang's user avatar
2 votes
0 answers
51 views

Generation function for the finite Coxeter group of type $D_k$

Basically I'd like to know how to derive the generation function for the finite Coxeter group of type $D_k$ to be familiar with notes in OEIS A162288: According to formula section: 'The growth series ...
Mikhail Gaichenkov's user avatar
1 vote
1 answer
92 views

Question about A051002 in OEIS

I have a question about sequence https://oeis.org/A051002 in the OEIS database with first elements 1, 1, 244, 1, 3126, 244. I searched however for ...
nilo de roock's user avatar
5 votes
1 answer
105 views

A recursive approach to Josephus problem (OEIS 225381)

Taking integer $x$ as input: If $x$ is even, then return $x/2$ as series item, giving $x/2$ as input to continue the loop; Else return ceiling($x/2$) as series item, terminate the loop; Finally we sum ...
viki's user avatar
  • 61
1 vote
0 answers
51 views

Series $\sum_{k=0}^\infty \frac{k^u}{k!}$---something known? [duplicate]

Define $f(u)=\sum_{k=0}^\infty \frac{k^u}{k!}$ for natural numbers $u$. Using maple I can find the sum for specific values of $u$, but there is no general answer (unsurprisingly). All values I found ...
kjetil b halvorsen's user avatar
1 vote
1 answer
165 views

Questions about the sequence OEIS A059650

Comparing the sequences OEIS A001951 $$a(n) = \lfloor n\sqrt{2}\rfloor = \lfloor \sqrt{2n^2 }\rfloor = \lfloor \sqrt{2*Square }\rfloor:\quad \{0, 1, 2, 4, 5, 7, 8, 9, 11, 12, 14, 15, 16, 18, 19, 21, .....
user avatar
2 votes
0 answers
57 views

Squares in the hypercube and humps in Motzkin paths

The number of squares with vertices coming from the vertices of the $n$-dimensional hypercube $\{0,1\}^n$ is given by $$ 2^{n-3}\sum_{j=1}^{n} \binom{n}{j}\binom{n-j}{j}. $$ The $0$-indexed sequence ...
Peter Kagey's user avatar
  • 5,062
0 votes
1 answer
80 views

Brun's theorem - notation

The series $1,3,6,9,12,14$ represent the first six terms of the series A168045 on OEIS. The formula for this says $a_n = 2n + n / log n + O(n / (log n)^2)$. I do not understand this notation. What ...
bissi's user avatar
  • 64
8 votes
1 answer
248 views

Octahedron to tetrahedron ratio in generalized polyominoes in the tetrahedral-octahedral honeycomb

Drake Thomas and I have proposed a sequence A343909 to the On-Line Encyclopedia of Integer Sequences (OEIS), which counts "generalized polyforms": generalizations of free polyominoes (Tetris ...
Peter Kagey's user avatar
  • 5,062