cp's OEIS Frontend

a(n+1) is the sum of the elements in the n-th row of triangle pertaining to A036561. - Amarnath Murthy, Jan 02 2002

Number of 2 X n binary arrays with a path of adjacent 1's and no path of adjacent 0's from top row to bottom row. - R. H. Hardin, Mar 21 2002

With offset 1, partial sums of A027649. - Paul Barry, Jun 24 2003

Number of distinct lines through the origin in the n-dimensional lattice of side length 2. A049691 has the values for the 2-dimensional lattice of side length n. - Joshua Zucker, Nov 19 2003

a(n+1)/(n+1)=(3*3^n-2*2^n)/(n+1) is the second binomial transform of the harmonic sequence 1/(n+1). - Paul Barry, Apr 19 2005

a(n+1) is the sum of n-th row of A036561. - Reinhard Zumkeller, May 14 2006

The sequence gives the sum of the lengths of the segments in Cantor's dust generating sequence up to the i-th step. Measurement unit = length of the segment of i-th step. - Giorgio Balzarotti, Nov 18 2006

Let T be a binary relation on the power set P(A) of a set A having n = |A| elements such that for every element x, y of P(A), xTy if x is a proper subset of y. Then a(n) = |T|. - Ross La Haye, Dec 22 2006

From Alexander Adamchuk, Jan 04 2007: (Start)

a(n) is prime for n in A057468.

p divides a(p) - 1 for prime p.

Quotients (3^p - 2^p - 1)/p, where p = prime(n), are listed in A127071.

Numbers k such that k divides 3^k - 2^k - 1 are listed in A127072.

Pseudoprimes in A127072(n) include all powers of primes {2,3,7} and some composite numbers that are listed in A127073, which includes all Carmichael numbers A002997.

Numbers n such that n^2 divides 3^n - 2^n - 1 are listed in A127074.

5 divides a(2n).

5^2 divides a(2*5n).

5^3 divides a(2*5^2n).

5^4 divides a(2*5^3n).

7^2 divides a(6*7n).

13 divides a(4n).

13^2 divides a(4*13n).

19 divides a(3n).

19^2 divides a(3*19n).

23^2 divides a(11n).

23^3 divides a(11*23n).

23^4 divides a(11*23^2n).

29 divides a(7n).

p divides a((p-1)n) for prime p>3.

p divides a((p-1)/2) for prime p in A097934. Also primes p such that 6 is a square mod p, except {2,3}, A038876(n).

p^(k+1) divides a(p^k*(p-1)/2*n) for prime p in A097934.

p^(k+1) divides a(p^k*(p-1)*n) for prime p>3.

Note the exception that for p = 23, p^(k+2) divides a(p^k*(p-1)/2*n).

There are no more such exceptions for primes p up to 600000. (End)

a(n) divides a(q*(n+1)-1), for all q integer. Leonardo Sarasua, Apr 15 2024

Final digits of terms follow sequence 1,5,9,5. - Enoch Haga, Nov 26 2007

This is also the second column sequence of the Sheffer triangle A143494 (2-restricted Stirling2 numbers). See the e.g.f. given below. - Wolfdieter Lang, Oct 08 2011

Partial sums give A000392. - Jon Perry, Apr 05 2014

For n >= 1, this is also row 2 of A281890: when consecutive positive integers are written as a product of primes in nondecreasing order, "3" occurs in n-th position a(n) times out of every 6^n. - Peter Munn, May 17 2017

a(n) is the number of ternary sequences of length n which include the digit 2. For example, a(2)=5 since the sequences are 02,20,12,21,22. - Enrique Navarrete, Apr 05 2021

a(n-1) is the number of ways we can form disjoint unions of two nonempty subsets of [n] such that the union contains n. For example, for n = 3, a(2) = 5 since the disjoint unions are {1}U{3}, {1}U{2,3}, {2}U{3}, {2}U{1,3}, and {1,2}U{3}. Cf. A000392 if we drop the requirement that the union contains n. - Enrique Navarrete, Aug 24 2021

Configures as a composite Koch Snowflake Fractal (see illustration in links) based on the five-fold division of the Cantor Square/Cantor Dust Fractal of (9^n-4^n)/5 see my illustration in (A016153). - John Elias, Oct 13 2021

Number of pairs (A,B) where B is a subset of {1,2,...,n} and A is a proper subset of B. - Jianing Song, Jun 18 2022

From Manfred Boergens, Mar 29 2023: (Start)

With regard to the comments by Ross La Haye and Jianing Song: Omitting "proper" gives A000244.

Number of pairs (A,B) where B is a nonempty subset of {1,2,...,n} and A is a nonempty subset of B. For nonempty proper subsets see a(n+1) in A028243. (End)

a(n) is the number of n-digit numbers whose smallest decimal digit is 7. - Stefano Spezia, Nov 15 2023

a(n-1) is the number of all possible player-reduced binary games observed by each player in an nx2 game assuming the individual strategies of k < n - 1 players are fixed and the remaining n - k - 1 player will play as one, either maintaining their status quo strategies or jointly adopting an alternative strategy. - Ambrosio Valencia-Romero, Apr 11 2024

Poly-Bernoulli numbers B_n^(k) with k=-2.

Binomial transform of A007051, if both sequences start at 0. Binomial transform of A000225(n+1). - Paul Barry, Mar 24 2003

Euler expands (1-z)/(1-5z+6z^2) and finds the general term. Section 226 of the Introductio indicates that he could have written down the recursion relation: a(n) = 5 a(n-1)-6 a(n-2). - V. Frederick Rickey (fred-rickey(AT)usma.edu), Feb 10 2006

Let R be a binary relation on the power set P(A) of a set A having n = |A| elements such that for every element x, y of P(A), xRy if x is a subset of y or y is a subset of x. Then a(n) = |R|. - Ross La Haye, Dec 22 2006

With regard to the comment by Ross La Haye: For proper subsets see A056182. - For nonempty subsets see A091344. - For nonempty proper subsets see a(n+1) in A260217. - Manfred Boergens, Aug 02 2023

If x, y are two n-bit binary strings then a(n) gives the number of pairs (x,y) such that XOR(x, y) = ABS(x - y). - Ramasamy Chandramouli, Feb 15 2009

Equals row sums of the triangular version of A038573. - Gary W. Adamson, Jun 04 2009

Inverse binomial transform of A085350. - Paul Curtz, Nov 14 2009

Related to the number of even a's in a nontrivial cycle (should one exist) in the 3x+1 Problem, where a <= floor(log_2(2*(3^n) - 2^n)). The value n correlates to the number of odds in such a nontrivial cycle. See page 1288 of Crandall's paper. Also, this relation gives another proof that the number of odds divided by the number of evens in a nontrivial cycle is bounded by log 2 / log 3 (this observation does not resolve the finite cycles conjecture as the value could be arbitrarily close to this bound). However, the same argument gives that log 2 / log 3 is less than or equal to the number of odds divided by the number of evens in a divergent sequence (should one exist), as log 2 / log 3 is the limit value for a cycle of an arbitrarily large length, where the length is given by the value n. - Jeffrey R. Goodwin, Aug 04 2011

Row sums of Riordan triangle A106516. - Wolfdieter Lang, Jan 09 2015

Number of restricted barred preferential arrangements having 3 bars in which the sections are all restricted sections such that (for fixed sections i and j) section i or section j is empty. - Sithembele Nkonkobe, Oct 12 2015

This is also row 2 of A281891: for n >= 1, when consecutive positive integers are written as a product of primes in nondecreasing order, a factor of 2 or 3 occurs in n-th position a(n) times out of every 6^n. - Peter Munn, May 18 2017

Also row sums of A124929. - Omar E. Pol, Jun 15 2017

This is the sum of A318921(n) for n in the range 2^(k+1) to 2^(k+2)-1. See A318921 for proof. - N. J. A. Sloane, Sep 25 2018

a(n) is also the number of acyclic orientations of the complete bipartite graph K_{2,n}. - Vincent Pilaud, Sep 15 2020

a(n-1) is also the number of n-digit numbers whose largest decimal digit is 2. - Stefano Spezia, Nov 15 2023

For n >= 3, a(n) is equal to the number of functions f: {1,2,...,n-1} -> {1,2,3} such that Im(f) contains 2 fixed elements. - Aleksandar M. Janjic and Milan Janjic, Mar 08 2007

Let P(A) be the power set of an n-element set A. Then a(n+1) = the number of pairs of elements {x,y} of P(A) for which x and y are intersecting and for which either x is a proper subset of y or y is a proper subset of x. - Ross La Haye, Jan 02 2008

Let P(A) be the power set of an n-element set A and R be a relation on P(A) such that for all x, y of P(A), xRy if x is not a subset of y and y is not a subset of x and x and y are disjoint. Then a(n+1) = |R|. - Ross La Haye, Mar 19 2009

Let P(A) be the power set of an n-element set A and R be a relation on P(A) such that for all x, y of P(A), xRy if either 0) x is a proper subset of y or y is a proper subset of x, or 1) x is not a subset of y and y is not a subset of x and x and y are disjoint. Then a(n+2) = |R|. - Ross La Haye, Mar 19 2009

In the terdragon curve, a(n) is the number of triple-visited points in expansion level n. The first differences of this sequence (A056182) are the number of enclosed unit triangles since on segment expansion each unit triangle forms a new triple-visited point, and existing triple-visited points are unchanged. - Kevin Ryde, Oct 20 2020

a(n+1) is the number of ternary strings of length n that contain at least one 0 and one 1; for example, for n=3, a(4)=12 since the strings are the 3 permutations of 100, the 3 permutations of 110, and the 6 permutations of 210. - Enrique Navarrete, Aug 13 2021

From Sanjay Ramassamy, Dec 23 2021: (Start)

a(n+1) is the number of topological configurations of n points and n lines where the points lie at the vertices of a convex cyclic n-gon and the lines are the perpendicular bisectors of its sides.

a(n+1) is the number of 2n-tuples composed of n 0's and n 1's which have an interlacing signature. The signature of a 2n-tuple (v_1,...,v_{2n}) is the n-tuple (s_1,...,s_n) defined by s_i=v_i+v_{i+n}. The signature is called interlacing if after deleting the 1's, there are letters remaining and the remaining 0's and 2's are alternating. (End)

a(n+1) is the number of pairs (A,B) where B is a nonempty subset of {1,2,...,n} and A is a nonempty proper subset of B. If either "nonempty" or "proper" is omitted then see A001047. If "nonempty" and "proper" are omitted then see A000244. - Manfred Boergens, Mar 28 2023

a(n) is the number of (n-1) X (n-1) nilpotent Boolean relation matrices with rank equal to 1. a(n) = A060867(n-1) - A005061(n-1) (since every rank 1 matrix is either idempotent or nilpotent). - Geoffrey Critzer, Jul 13 2023

For odd n > 3, a(n) is also the number of minimum vertex colorings in the (n-1)-prism graph. - Eric W. Weisstein, Mar 05 2024

Binomial transform of A000918: (-1, 0, 2, 6, 14, 30, ...). - Gary W. Adamson, Mar 23 2012

This sequence demonstrates 2^n as a loose lower bound for g(n) in Waring's problem. Since 3^n > 2(2^n) for all n > 2, the number 2^(n + 1) - 1 requires 2^n n-th powers for its representation since 3^n is not available for use in the sum: the gulf between the relevant powers of 2 and 3 widens considerably as n gets progressively larger. - Alonso del Arte, Feb 01 2013

See A002872 for detailed description.

T(m,k) is the number of achiral color patterns in a row or loop of length 2m using exactly k different colors. Two color patterns are equivalent if we permute the colors. - Robert A. Russell, Apr 24 2018

T(n,k) = coefficient of x^k for A(2,n)(x) in Gilbert and Riordan's article. - Robert A. Russell, Jun 14 2018

Starting with offset 1 = binomial transform of A068293: (1, 6, 18, 42, 90, ...) and double binomial transform of (1, 5, 7, 5, 7, 5, ...). - Gary W. Adamson, Jan 13 2009

Number of pairs (A,B) where A and B are nonempty subsets of {1,2,...,n} and one of these subsets is a subset of the other. - For the case that one of these subsets is a proper subset of the other see a(n+1) in A260217. - If empty subsets are included, see A027649 (all subsets) and A056182 (proper subsets). - Manfred Boergens, Aug 02 2023

Columns 0,1,2,3 respectively correspond to relations R_3, R_4, R_0, R_1 defined in La Haye paper listed below.

T(n,k) is the number of permutations p of [n] such that max(p(i) - i) = k. Example: T(3,0) = 1 because for p = 123 we have max(p(i) - i) = 0; T(3,1) = 3 because for p = 132, 213, 231 we have max(p(i) - i) = 1; T(3,2) = 2 because for p = 312, 321 we have max(p(i) - i) = 2. - Emeric Deutsch, Nov 12 2004

T(n,k) is the number of permutations of [n] for which the first subcedance occurs at position n + 2 - k. A subcedance of pi occurs at position i if i>pi(i). For example, with n = 3 and k = 2, T(n,k) = 3 counts 132, 231, 321 in each of which the first subcedance occurs at position n+2-k = 3. - David Callan, Dec 14 2021

Column 1: -1+2^n.

Row sums: A048473.

Alternating row sums: 1,1,1,1,1,1,1,1,1,...

For a discussion and guide to related arrays, see A208510.

Row sums without first column give A056182. - Alois P. Heinz, Jan 14 2022