cp's OEIS Frontend

From Lara Pudwell, Oct 23 2008: (Start)

A permutation p avoids a pattern q if it has no subsequence that is order-isomorphic to q. For example, p avoids the pattern 132 if it has no subsequence abc with a < c < b.

Barred pattern avoidance considers permutations that avoid a pattern except in a special case. Given a barred pattern q, we may form two patterns, q1 = the sequence of unbarred letters of q and q2 = the sequence of all letters of q.

A permutation p avoids barred pattern q if every instance of q1 in p is embedded in a copy of q2 in p. In other words, p avoids q1, except in the special case that a copy of q1 is a subsequence of a copy of q2.

For example, if q=5{bar 1}32{bar 4}, then q1=532 and q2 = 51324. p avoids q if every for decreasing subsequence acd of length 3 in p, one can find letters b and e so that the subsequence abcde of p has b < d < c < e < a. (End)

Number of ordered factorizations over the Gaussian polynomials.

Apparently, also the number of permutations in S_n avoiding {bar 3}{bar 1}542 (i.e., every occurrence of 542 is contained in an occurrence of a 31542). - Lara Pudwell, Apr 25 2008

With offset 1, apparently the number of sequences {b(m)} of length n of positive integers with b(1) = 1 and, for all m > 1, b(m) <= max{b(m-1) + 1, max{b(i) | 1 <= i <= m - 1}}. This sequence begins 1, 2, 5, 14, 43, 144, 523, 2048, 8597, 38486. The term 144 counts the length 6 sequence 1, 2, 3, 1, 1, 3, for instance. Contrast with the families of sequences discussed in Franklin T. Adams-Watters's comment in A005425. - Rick L. Shepherd, Jan 01 2015

a(n-1) for n >= 1 is the number of length-n restricted growth strings (RGS) [s(0), s(1), ..., s(n-1)] with s(0)=0 and s(k) <= the number of fixed points in the prefix, see example. - Joerg Arndt, Mar 08 2015

Number of sequences (e(1), ..., e(n+1)), 0 <= e(i) < i, such that there is no triple i < j < k with e(i) != e(j) = e(k). [Martinez and Savage, 2.15] - Eric M. Schmidt, Jul 17 2017

a(n) counts all positive-integer m-tuples whose maximum is n-m+2. - Mathew Englander, Feb 28 2021

a(n) counts the cyclic permutations of [n+2] that avoid the vincular pattern 12-3-4, i.e., the pattern 1234 where the 1 and 2 are required to be adjacent. - Rupert Li, Jul 27 2021

The indexing used here follows that given in the classic books by Riordan and Comtet. For two other versions see A173018 and A123125. - N. J. A. Sloane, Nov 21 2010

Coefficients of Eulerian polynomials. Number of permutations of n objects with k-1 rises. Number of increasing rooted trees with n+1 nodes and k leaves.

T(n,k) = number of permutations of [n] with k runs. T(n,k) = number of permutations of [n] requiring k readings (see the Knuth reference). T(n,k) = number of permutations of [n] having k distinct entries in its inversion table. - Emeric Deutsch, Jun 09 2004

T(n,k) = number of ways to write the Coxeter element s_{e1}s_{e1-e2}s_{e2-e3}s_{e3-e4}...s_{e_{n-1}-e_n} of the reflection group of type B_n, using s_{e_k} and as few reflections of the form s_{e_i+e_j}, where i = 1, 2, ..., n and j is not equal to i, as possible. - Pramook Khungurn (pramook(AT)mit.edu), Jul 07 2004

Subtriangle for k>=1 and n>=1 of triangle A123125. - Philippe Deléham, Oct 22 2006

T(n,k)/n! also represents the n-dimensional volume of the portion of the n-dimensional hypercube cut by the (n-1)-dimensional hyperplanes x_1 + x_2 + ... x_n = k, x_1 + x_2 + ... x_n = k-1; or, equivalently, it represents the probability that the sum of n independent random variables with uniform distribution between 0 and 1 is between k-1 and k. - Stefano Zunino, Oct 25 2006

[E(.,t)/(1-t)]^n = n!*Lag[n,-P(.,t)/(1-t)] and [-P(.,t)/(1-t)]^n = n!*Lag[n, E(.,t)/(1-t)] umbrally comprise a combinatorial Laguerre transform pair, where E(n,t) are the Eulerian polynomials and P(n,t) are the polynomials in A131758. - Tom Copeland, Sep 30 2007

From Tom Copeland, Oct 07 2008: (Start)

G(x,t) = 1/(1 + (1-exp(x*t))/t) = 1 + 1*x + (2+t)*x^2/2! + (6+6*t+t^2)*x^3/3! + ... gives row polynomials for A090582, the reverse f-polynomials for the permutohedra (see A019538).

G(x,t-1) = 1 + 1*x + (1+t)*x^2/2! + (1+4*t+t^2)*x^3/3! + ... gives row polynomials for A008292, the h-polynomials for permutohedra (Postnikov et al.).

G((t+1)*x, -1/(t+1)) = 1 + (1+t)*x + (1+3*t+2*t^2)*x^2/2! + ... gives row polynomials for A028246.

(End)

A subexceedant function f on [n] is a map f:[n] -> [n] such that 1 <= f(i) <= i for all i, 1 <= i <= n. T(n,k) equals the number of subexceedant functions f of [n] such that the image of f has cardinality k [Mantaci & Rakotondrajao]. Example T(3,2) = 4: if we identify a subexceedant function f with the word f(1)f(2)...f(n) then the subexceedant functions on [3] are 111, 112, 113, 121, 122 and 123 and four of these functions have an image set of cardinality 2. - Peter Bala, Oct 21 2008

Further to the comments of Tom Copeland above, the n-th row of this triangle is the h-vector of the simplicial complex dual to a permutohedron of type A_(n-1). The corresponding f-vectors are the rows of A019538. For example, 1 + 4*x + x^2 = y^2 + 6*y + 6 and 1 + 11*x + 11*x^2 + x^3 = y^3 + 14*y^2 + 36*y + 24, where x = y + 1, give [1,6,6] and [1,14,36,24] as the third and fourth rows of A019538. The Hilbert transform of this triangle (see A145905 for the definition) is A047969. See A060187 for the triangle of Eulerian numbers of type B (the h-vectors of the simplicial complexes dual to permutohedra of type B). See A066094 for the array of h-vectors of type D. For tables of restricted Eulerian numbers see A144696 - A144699. - Peter Bala, Oct 26 2008

For a natural refinement of A008292 with connections to compositional inversion and iterated derivatives, see A145271. - Tom Copeland, Nov 06 2008

The polynomials E(z,n) = numerator(Sum_{k>=1} (-1)^(n+1)*k^n*z^(k-1)) for n >=1 lead directly to the triangle of Eulerian numbers. - Johannes W. Meijer, May 24 2009

From Walther Janous (walther.janous(AT)tirol.com), Nov 01 2009: (Start)

The (Eulerian) polynomials e(n,x) = Sum_{k=0..n-1} T(n,k+1)*x^k turn out to be also the numerators of the closed-form expressions of the infinite sums:

S(p,x) = Sum_{j>=0} (j+1)^p*x^j, that is

S(p,x) = e(p,x)/(1-x)^(p+1), whenever |x| < 1 and p is a positive integer.

(Note the inconsistent use of T(n,k) in the section listing the formula section. I adhere tacitly to the first one.) (End)

If n is an odd prime, then all numbers of the (n-2)-th and (n-1)-th rows are in the progression k*n+1. - Vladimir Shevelev, Jul 01 2011

The Eulerian triangle is an element of the formula for the r-th successive summation of Sum_{k=1..n} k^j which appears to be Sum_{k=1..n} T(j,k-1) * binomial(j-k+n+r, j+r). - Gary Detlefs, Nov 11 2011

Li and Wong show that T(n,k) counts the combinatorially inequivalent star polygons with n+1 vertices and sum of angles (2*k-n-1)*Pi. An equivalent formulation is: define the total sign change S(p) of a permutation p in the symmetric group S_n to be equal to Sum_{i=1..n} sign(p(i)-p(i+1)), where we take p(n+1) = p(1). T(n,k) gives the number of permutations q in S_(n+1) with q(1) = 1 and S(q) = 2*k-n-1. For example, T(3,2) = 4 since in S_4 the permutations (1243), (1324), (1342) and (1423) have total sign change 0. - Peter Bala, Dec 27 2011

Xiong, Hall and Tsao refer to Riordan and mention that a traditional Eulerian number A(n,k) is the number of permutations of (1,2...n) with k weak exceedances. - Susanne Wienand, Aug 25 2014

Connections to algebraic geometry/topology and characteristic classes are discussed in the Buchstaber and Bunkova, the Copeland, the Hirzebruch, the Lenart and Zainoulline, the Losev and Manin, and the Sheppeard links; to the Grassmannian, in the Copeland, the Farber and Postnikov, the Sheppeard, and the Williams links; and to compositional inversion and differential operators, in the Copeland and the Parker links. - Tom Copeland, Oct 20 2015

The bivariate e.g.f. noted in the formulas is related to multiplying edges in certain graphs discussed in the Aluffi-Marcolli link. See p. 42. - Tom Copeland, Dec 18 2016

Distribution of left children in treeshelves is given by a shift of the Eulerian numbers. Treeshelves are ordered binary (0-1-2) increasing trees where every child is connected to its parent by a left or a right link. See A278677, A278678 or A278679 for more definitions and examples. - Sergey Kirgizov, Dec 24 2016

The row polynomial P(n, x) = Sum_{k=1..n} T(n, k)*x^k appears in the numerator of the o.g.f. G(n, x) = Sum_{m>=0} S(n, m)*x^m with S(n, m) = Sum_{j=0..m} j^n for n >= 1 as G(n, x) = Sum_{k=1..n} P(n, x)/(1 - x)^(n+2) for n >= 0 (with 0^0=1). See also triangle A131689 with a Mar 31 2017 comment for a rewritten form. For the e.g.f see A028246 with a Mar 13 2017 comment. - Wolfdieter Lang, Mar 31 2017

For relations to Ehrhart polynomials, volumes of polytopes, polylogarithms, the Todd operator, and other special functions, polynomials, and sequences, see A131758 and the references therein. - Tom Copeland, Jun 20 2017

For relations to values of the Riemann zeta function at integral arguments, see A131758 and the Dupont reference. - Tom Copeland, Mar 19 2018

Normalized volumes of the hypersimplices, attributed to Laplace. (Cf. the De Loera et al. reference, p. 327.) - Tom Copeland, Jun 25 2018

The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.

Crystal ball sequence for A_2 lattice. - Michael Somos, Jun 03 2012

Sixth spoke of hexagonal spiral (cf. A056105-A056109).

Number of ordered integer triples (a,b,c), -n <= a,b,c <= n, such that a+b+c=0. - Benoit Cloitre, Jun 14 2003

Also the number of partitions of 6n into at most 3 parts, A001399(6n). - R. K. Guy, Oct 20 2003

Also, a(n) is the number of partitions of 6(n+1) into exactly 3 distinct parts. - William J. Keith, Jul 01 2004

Number of dots in a centered hexagonal figure with n+1 dots on each side.

Values of second Bessel polynomial y_2(n) (see A001498).

First differences of cubes (A000578). - Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 15 2004

Final digits of Hex numbers (hex(n) mod 10) are periodic with palindromic period of length 5 {1, 7, 9, 7, 1}. Last two digits of Hex numbers (hex(n) mod 100) are periodic with palindromic period of length 100. - Alexander Adamchuk, Aug 11 2006

All divisors of a(n) are congruent to 1, modulo 6. Proof: If p is an odd prime different from 3 then 3n^2 + 3n + 1 = 0 (mod p) implies 9(2n + 1)^2 = -3 (mod p), whence p = 1 (mod 6). - Nick Hobson, Nov 13 2006

For n>=1, a(n) is the side of Outer Napoleon Triangle whose reference triangle is a right triangle with legs (3a(n))^(1/2) and 3n(a(n))^(1/2). - Tom Schicker (tschicke(AT)email.smith.edu), Apr 25 2007

Number of triples (a,b,c) where 0<=(a,b)<=n and c=n (at least once the term n). E.g., for n = 1: (0,0,1), (0,1,0), (1,0,0), (0,1,1), (1,0,1), (1,1,0), (1,1,1), so a(1)=7. - Philippe Lallouet (philip.lallouet(AT)wanadoo.fr), Aug 20 2007

Equals the triangular numbers convolved with [1, 4, 1, 0, 0, 0, ...]. - Gary W. Adamson and Alexander R. Povolotsky, May 29 2009

From Terry Stickels, Dec 07 2009: (Start)

Also the maximum number of viewable cubes from any one static point while viewing a cube stack of identical cubes of varying magnitude.

For example, viewing a 2 X 2 X 2 stack will yield 7 maximum viewable cubes.

If the stack is 3 X 3 X 3, the maximum number of viewable cubes from any one static position is 19, and so on.

The number of cubes in the stack must always be the same number for width, length, height (at true regular cubic stack) and the maximum number of visible cubes can always be found by taking any cubic number and subtracting the number of the cube that is one less.

Examples: 125 - 64 = 61, 64 - 27 = 37, 27 - 8 = 19. (End)

The sequence of digital roots of the a(n) is period 3: repeat [1,7,1]. - Ant King, Jun 17 2012

The average of the first n (n>0) centered hexagonal numbers is the n-th square. - Philippe Deléham, Feb 04 2013

A002024 is the following array A read along antidiagonals:

1, 2, 3, 4, 5, 6, ...

2, 3, 4, 5, 6, 7, ...

3, 4, 5, 6, 7, 8, ...

4, 5, 6, 7, 8, 9, ...

5, 6, 7, 8, 9, 10, ...

6, 7, 8, 9, 10, 11, ...

and a(n) is the hook sum Sum_{k=0..n} A(n,k) + Sum_{r=0..n-1} A(r,n). - R. J. Mathar, Jun 30 2013

a(n) is the sum of the terms in the n+1 X n+1 matrices minus those in n X n matrices in an array formed by considering A158405 an array (the beginning terms in each row are 1,3,5,7,9,11,...). - J. M. Bergot, Jul 05 2013

The formula also equals the product of the three distinct combinations of two consecutive numbers: n^2, (n+1)^2, and n*(n+1). - J. M. Bergot, Mar 28 2014

The sides of any triangle ABC are divided into 2n + 1 equal segments by 2n points: A_1, A_2, ..., A_2n in side a, and also on the sides b and c cyclically. If A'B'C' is the triangle delimited by AA_n, BB_n and CC_n cevians, we have (ABC)/(A'B'C') = a(n) (see Java applet link). - Ignacio Larrosa Cañestro, Jan 02 2015

a(n) is the maximal number of parts into which (n+1) triangles can intersect one another. - Ivan N. Ianakiev, Feb 18 2015

((2^m-1)n)^t mod a(n) = ((2^m-1)(n+1))^t mod a(n) = ((2^m-1)(2n+1))^t mod a(n), where m any positive integer, and t = 0(mod 6). - Alzhekeyev Ascar M, Oct 07 2016

((2^m-1)n)^t mod a(n) = ((2^m-1)(n+1))^t mod a(n) = a(n) - (((2^m-1)(2n+1))^t mod a(n)), where m any positive integer, and t = 3(mod 6). - Alzhekeyev Ascar M, Oct 07 2016

(3n+1)^(a(n)-1) mod a(n) = (3n+2)^(a(n)-1) mod a(n) = 1. If a(n) not prime, then always strong pseudoprime. - Alzhekeyev Ascar M, Oct 07 2016

Every positive integer is the sum of 8 hex numbers (zero included), at most 3 of which are greater than 1. - Mauro Fiorentini, Jan 01 2018

Area enclosed by the segment of Archimedean spiral between n*Pi/2 and (n+1)*Pi/2 in Pi^3/48 units. - Carmine Suriano, Apr 10 2018

This sequence contains all numbers k such that 12*k - 3 is a square. - Klaus Purath, Oct 19 2021

The continued fraction expansion of sqrt(3*a(n)) is [3n+1; {1, 1, 2n, 1, 1, 6n+2}]. For n = 0, this collapses to [1; {1, 2}]. - Magus K. Chu, Sep 12 2022

Number of ways n labeled objects can be distributed into k nonempty parcels. Also number of special terms in n variables with maximal degree k.

In older terminology these are called differences of 0. - Michael Somos, Oct 08 2003

Number of surjections (onto functions) from an n-element set to a k-element set.

Also coefficients (in ascending order) of so-called ordered Bell polynomials.

(k-1)!*Stirling2(n,k-1) is the number of chain topologies on an n-set having k open sets [Stephen].

Number of set compositions (ordered set partitions) of n items into k parts. Number of k dimensional 'faces' of the n dimensional permutohedron (see Simion, p. 162). - Mitch Harris, Jan 16 2007

Correction of comment before: Number of (n-k)-dimensional 'faces' of the permutohedron of order n (an (n-1)-dimensional polytope). - Tilman Piesk, Oct 29 2014

This array is related to the reciprocal of an e.g.f. as sketched in A133314. For example, the coefficient of the fourth-order term in the Taylor series expansion of 1/(a(0) + a(1) x + a(2) x^2/2! + a(3) x^3/3! + ...) is a(0)^(-5) * {24 a(1)^4 - 36 a(1)^2 a(2) a(0) + [8 a(1) a(3) + 6 a(2)^2] a(0)^2 - a(4) a(0)^3}. The unsigned coefficients characterize the P3 permutohedron depicted on page 10 in the Loday link with 24 vertices (0-D faces), 36 edges (1-D faces), 6 squares (2-D faces), 8 hexagons (2-D faces) and 1 3-D permutohedron. Summing coefficients over like dimensions gives A019538 and A090582. Compare to A133437 for the associahedron. - Tom Copeland, Sep 29 2008, Oct 07 2008

Further to the comments of Tom Copeland above, the permutohedron of type A_3 can be taken as the truncated octahedron. Its dual is the tetrakis hexahedron, a simplicial polyhedron, with f-vector (1,14,36,24) giving the fourth row of this triangle. See the Wikipedia entry and [Fomin and Reading p. 21]. The corresponding h-vectors of permutohedra of type A give the rows of the triangle of Eulerian numbers A008292. See A145901 and A145902 for the array of f-vectors for type B and type D permutohedra respectively. - Peter Bala, Oct 26 2008

Subtriangle of triangle in A131689. - Philippe Deléham, Nov 03 2008

Since T(n,k) counts surjective functions and surjective functions are "consistent", T(n,k) satisfies a binomial identity, namely, T(n,x+y) = Sum_{j=0..n} C(n,j)*T(j,x)*T(n-j,y). For definition of consistent functions and a generalized binomial identity, see "Toy stories and combinatorial identities" in the link section below. - Dennis P. Walsh, Feb 24 2012

T(n,k) is the number of labeled forests on n+k vertices satisfying the following two conditions: (i) each forest consists of exactly k rooted trees with roots labeled 1, 2, ..., k; (ii) every root has at least one child vertex. - Dennis P. Walsh, Feb 24 2012

The triangle is the inverse binomial transform of triangle A028246, deleting the left column and shifting up one row. - Gary W. Adamson, Mar 05 2012

See A074909 for associations among this array and the Bernoulli polynomials and their umbral compositional inverses. - Tom Copeland, Nov 14 2014

E.g.f. for the shifted signed polynomials is G(x,t) = (e^t-1)/[1+(1+x)(e^t-1)] = 1-(1+x)(e^t-1) + (1+x)^2(e^t-1)^2 - ... (see also A008292 and A074909), which has the infinitesimal generator g(x,u)d/du = [(1-x*u)(1-(1+x)u)]d/du, i.e., exp[t*g(x,u)d/du]u eval. at u=0 gives G(x,t), and dG(x,t)/dt = g(x,G(x,t)). The compositional inverse is log((1-xt)/(1-(1+x)t)). G(x,t) is a generating series associated to the generalized Hirzebruch genera. See the G. Rzadowski link for the relation of the derivatives of g(x,u) to solutions of the Riccatt differential equation, soliton solns. to the KdV equation, and the Eulerian and Bernoulli numbers. In addition A145271 connects products of derivatives of g(x,u) and the refined Eulerian numbers to the inverse of G(x,t), which gives the normalized, reverse face polynomials of the simplices (A135278, divided by n+1). See A028246 for the generator g(x,u)d/dx. - Tom Copeland, Nov 21 2014

For connections to toric varieties and Eulerian polynomials, see the Dolgachev and Lunts and the Stembridge links. - Tom Copeland, Dec 31 2015

See A008279 for a relation between the e.g.f.s enumerating the faces of permutahedra (this entry) and stellahedra. - Tom Copeland, Nov 14 2016

T(n, k) appears in a Worpitzky identity relating monomials to binomials: x^n = Sum_{k=1..n} T(n, k)*binomial(x,k), n >= 1. See eq. (11.) of the Worpitzky link on p. 209. The relation to the Eulerian numbers is given there in eqs. (14.) and (15.). See the formula below relating to A008292. See also Graham et al. eq. (6.10) (relating monomials to falling factorials) on p. 248 (2nd ed. p. 262). The Worpitzky identity given in the Graham et al. reference as eq. (6.37) (2nd ed. p. 269) is eq. (5.), p. 207, of Worpitzky. - Wolfdieter Lang, Mar 10 2017

T(n, m) is also the number of minimum clique coverings and minimum matchings in the complete bipartite graph K_{m,n}. - Eric W. Weisstein, Apr 26 2017

From the Hasan and Franco and Hasan papers: The m-permutohedra for m=1,2,3,4 are the line segment, hexagon, truncated octahedron and omnitruncated 5-cell. The first three are well-known from the study of elliptic models, brane tilings and brane brick models. The m+1 torus can be tiled by a single (m+2)-permutohedron. Relations to toric Calabi-Yau Kahler manifolds are also discussed. - Tom Copeland, May 14 2020

From Manfred Boergens, Jul 25 2021: (Start)

Number of n X k binary matrices with row sums = 1 and no zero columns. These matrices are a subset of the matrices defining A183109.

The distribution into parcels in the leading comment can be regarded as a covering of [n] by tuples (A_1,...,A_k) in P([n])^k with nonempty and disjoint A_j, with P(.) denoting the power set (corrected for clarity by Manfred Boergens, May 26 2024). For the non-disjoint case see A183109 and A218695.

For tuples with "nonempty" dropped see A089072. For tuples with "nonempty and disjoint" dropped see A092477 and A329943 (amendment by Manfred Boergens, Jun 24 2024). (End)

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

Number of 2 X n binary arrays with a path of adjacent 1's from top row to bottom row, see A359576. - R. H. Hardin, Mar 21 2002

Number of binary vectors (x_1, x_2, ..., x_{2n}) such that in at least one of the disjoint pairs (x_1, x_2), (x_3, x_4), ..., (x_{2n-1}, x_{2n}) both x_{2i-1} and x_{2i} are both 1. Equivalently, number of solutions (x_1, ..., x_n) to the equation x_1*x_2 + x_3*x_4 + x_5*x_6 + ... +x_{2n-1}*x_{2n} = 1 in base-2 lunar arithmetic. - N. J. A. Sloane, Apr 23 2011

a(n)/4^n is the probability that two randomly selected (with replacement) subsets of [n] will have at least one element in common if the probability of selection is equal for all subsets. - Geoffrey Critzer, May 09 2009

This sequence is also the second column of the Sheffer triangle A143495 (3-restricted Stirling2 numbers). (See the e.g.f. given below.) - Wolfdieter Lang, Oct 08 2011

Also, the number of numbers with at most n digits whose largest digit equals 3. See A255463 for the first differences (i.e., ...with exactly n digits...). - M. F. Hasler, May 03 2015

If 2^k | n then a(2^k) | a(n). - Bernard Schott, Oct 08 2020

a(n) is the number of ordered n-tuples with elements from {0,1,2,3} in which any of these elements, say 0, appears at least once. For example, a(2)=7 since 01,10,02,20,03,30,00 are the ordered 2-tuples that contain 0. - Enrique Navarrete, Apr 05 2021

a(n) is the number of n-digit numbers whose smallest decimal digit is 6. - Stefano Spezia, Nov 15 2023

Final digits of a(n), i.e., a(n) mod 10, are repeated periodically with period of length 5 {1,5,5,5,9}. There is a symmetry in this list since the sum of two numbers equally distant from the ends is equal to 10 = 1 + 9 = 5 + 5 = 2*5. Last two digits of a(n), i.e., a(n) mod 100, are repeated periodically with period of length 50. - Alexander Adamchuk, Aug 11 2006

a(n) = VarScheme(n,2) in the scheme displayed in A128195. - Peter Luschny, Feb 26 2007

If Y is a 3-subset of a 2n-set X then, for n >= 2, a(n-2) is the number of 4-subsets of X intersecting Y. - Milan Janjic, Nov 18 2007

The numbers are the constant number found in magic squares of order n, where n is an odd number, see the comment in A006003. A Magic Square of side 1 is 1; 3 is 15; 5 is 65 and so on. - David Quentin Dauthier, Nov 07 2008

Two times the area of the triangle with vertices at (0,0), ((n - 1)^2, n^2), and (n^2, (n - 1)^2). - J. M. Bergot, Jun 25 2013

Bisection of A006003. - Omar E. Pol, Sep 01 2018

Construct an array M with M(0,n) = 2*n^2 + 4*n + 1 = A056220(n+1), M(n,0) = 2*n^2 + 1 = A058331(n) and M(n,n) = 2*n*(n+1) + 1 = A001844(n). Row(n) begins with all the increasing odd numbers from A058331(n) to A001844(n) and column(n) begins with all the decreasing odd numbers from A056220(n+1) to A001844(n). The sum of the terms in row(n) plus those in column(n) minus M(n,n) equals a(n+1). The first five rows of array M are [1, 7, 17, 31, 49, ...]; [3, 5, 15, 29, 47, ...]; [9, 11, 13, 27, 45, ...]; [19, 21, 23, 25, 43, ...]; [33, 35, 37, 39, 41, ...]. - J. M. Bergot, Jul 16 2013 [This contribution was moved here from A047926 by Petros Hadjicostas, Mar 08 2021.]

For n>=2, these are the primitive sides s of squares of type 2 described in A344332. - Bernard Schott, Jun 04 2021

(a(n) + 1) / 2 = A212133(n) is the number of cells in the n-th rhombic-dodecahedral polycube. - George Sicherman, Jan 21 2024