cp's OEIS Frontend

a(n+1) is the number of 2 X n binary matrices with no zero rows or columns, up to row and column permutation.

[ (4th elementary symmetric function of S(n))/(3rd elementary symmetric function of S(n)) ], where S(n) = {first n+3 odd positive integers}.

First differences are 1, 2, 2, 3, 3, 4, 4, 5, 5, ... .

Let M_n denotes the n X n matrix m(i,j) = 1 if i =j; m(i,j) = 1 if (i+j) is odd; m(i,j) = 0 if i+j is even, then a(n) = -det M_(n+1) - Benoit Cloitre, Jun 19 2002

a(n) is the number of squares with corners on an n X n grid, distinct up to translation. See also A002415, A108279.

Starting (1, 3, 5, 8, 11, ...), = row sums of triangle A135841. - Gary W. Adamson, Dec 01 2007

Number of solutions to x+y >= n-1 in integers x,y with 1 <= x <= y <= n-1. - Franz Vrabec, Feb 22 2008

Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=-1, A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=5, a(n-4)=-coeff(charpoly(A,x),x^2). - Milan Janjic, Jan 26 2010

Equals row sums of a triangle with alternate columns of (1,2,3,...) and (1,1,1,...). - Gary W. Adamson, May 21 2010

Conjecture: if a(n) = p#(primorial)-1 for some prime number p, then q=(n+1) is also a prime number where p#=floor(q^2/4). Tested up to n=10^100000 no counterexamples are found. It seems that the subsequence is very scattered. So far the triples (p,q,a(q-1)) are {(2,3,1), (3,5,5), (5,11,29), (7,29,209), (17,1429,510509)}. - David Morales Marciel, Oct 02 2015

Numbers of an Ulam spiral starting at 0 in which the shape of the spiral is exactly a rectangle. E.g., a(4)=5 the Ulam spiral is including at that moment only the elements 0,1,2,3,4,5 and the shape is a rectangle. The area is always a(n)+1. E.g., for a(4) the area of the rectangle is 2(rows) X 3(columns) = 6 = a(4) + 1. - David Morales Marciel, Apr 05 2016

Numbers of different quadratic forms (quadrics) in the real projective space P^n(R). - Serkan Sonel, Aug 26 2020

a(n+1) is the number of one-dimensional subspaces of (F_3)^n, counted up to coordinate permutation. E.g.: For n=4, there are five one-dimensional subspaces in (F_3)^3 up to coordinate permutation: [1 2 2] [0 2 2] [1 0 2] [0 0 2] [1 1 1]. This example suggests a bijection (which has to be adjusted for the all-ones matrix) with the binary matrices of the first comment. - Álvar Ibeas, Sep 21 2021

Apparently, beginning with a(3), number of non-equivalent canonical forms of separation coordinates on the hyperspheres. Cf. Schöbel and Veselov for this and other interpretations. - Tom Copeland, Nov 21 2017

From Petros Hadjicostas, Jan 17 2018: (Start)

Let A(x) = Sum_{n>=1} a(n)*x^n. For a derivation of the formula BIK(A(x)) = (1/2)*(A(x)/(1-A(x)) + (A(x) + A(x^2))/(1-A(x^2))), see the comments for sequence A001224 and the weblink below containing Bower's theory of transforms.

We clarify the comment by T. Copeland above. Consider the material in Section 3 of Devadoss and Read (2001). According to their terminology, let b(m,n) be "the number of A-clusters having m cells and n outside edges not counting the root edge." Let B(x,y) = Sum_{m>=0, n>=0} b(m,n)*x^m*y^n. (See p. 78 in their paper, where they use the notations a_{m,n} and A(x,y) rather than b(m,n) and B(x,y), respectively, that we use here.)

On p. 79 (Eq. (3.1)) of their paper, they prove that B(x,y) = y + (x/2)*(B(x,y)^2/(1-B(x,y)) + (1 + B(x,y))*B(x^2, y^2)/(1-B(x^2,y^2))). Unfortunately, the factor x in the previous formula is left out (i.e., it is a typo), and the same typo is reproduced in Schöbel and Veselov (2014, 2015).

Using Table 2 (p. 92) from Devadoss and Read (2001) (and the material on p. 79), we get that B(x,y) = y+ x*y^2 + (x^2 + x)*y^3 + (2*x^3 + 3*x^2 + x)*y^4 + (3*x^4 + 8*x^3 + 5*x^2 + x)*y^5 + ...

We claim that a(n) = Sum_{m>=0} b(m,n) and A(y) = Sum_{n>=1} a(n)*y^n = B(x=1, y). To prove these claims, note that, for x=1, the above series becomes B(x=1,y) = y + y^2 + 2*y^3 + 6*y^4 + 17*y^5 + ..., while the functional equation above becomes B(1, y) = y + (1/2)*(B(1,y)^2/(1-B(1,y)) + (1 + B(1,y))*B(1,y^2)/(1-B(1,y^2))), which is equivalent to 2*B(1,y) = y + (1/2)*(B(1,y)/(1-B(1,y)) + (B(1,y) + B(1,y^2))/(1-B(1,y^2))). The latter formula is the one given in the formula section below (derived from Bower's theory) with x replaced with y and A(x) replaced with B(1,y). This proves that B(x=1, y) = A(y), from which we can easily get that a(n) = Sum_{m>=0} b(m,n).

Note that b(m=0, n) = 0 for n <> 1, but b(m=0, n=1) = 1; b(m,n) = 0 when m >= n >= 1; and b(m=1, n) = 1 for n>=2. Also, b(m,m+1) = A001190(m+1) for m>=1, which are the Wedderburn-Etherington numbers, and apparently b(m=2, n) = A024206(n-1) for n>=2 (conjecture).

In Section 6 of their paper, Schöbel and Veselov (2014, 2015) prove that b(m,n) is the "number of non-equivalent faces of [the Stasheff polytope] K_n of codimension m-1." Apparently then, for n>=2 and k>=0, b(n-k,n+1) is the "number of canonical forms for separation coordinates of [hypersphere] S^n" with k "independent continuous parameters". For k=0 and n>=2, b(n,n+1) = A001190(n+1) = "number of canonical forms for separation coordinates" of hypersphere S^n with 0 continuous parameters.

It turns out that for k, the number of continuous parameters of S^n, we have 0 <= k <= n-1 (see pp. 1269-1270 in Shobel and Veselov (2015)). Hence, for n>=2, Sum_{k=0..n-1} b(n-k, n+1) = Sum_{m=1..n} b(m, n+1) = Sum_{m=0..n} b(m, n+1) = a(n+1) (see above). As a result, for n>=2, a(n+1) is the "total number of [non-equivalent] canonical forms for separation coordinates on [hypersphere] S^n", which is the comment made by T. Copeland above.

(End)

For an explanation on the meaning of clusters of types A, B, and C see Section 3 (pp. 78-81) in Devadoos and Read (2001). See also the comments for sequence A232206. - Petros Hadjicostas, Mar 02 2018

Table 1 of the Schöbel and Veselov paper with initial 1 added. Reverse of Table 2 of the Devadoss and Read paper.

Apparently A032132 contains the row sums.

From Petros Hadjicostas, Jan 28 2018: (Start)

In this triangle, which is read by rows, for 0 <= k <= n-1 and n>=1, let T(n,k) be the number of inequivalent canonical forms for separation coordinates of the hypersphere S^n with k independent continuous parameters. It is the mirror image of sequence A232206, that is, T(n, k) = A232206(n+1, n-k) for 0 <= k <= n-1 and n>=1. (Triangular array A232206(N, K) is defined for N >= 2 and 1 <= K <= N-1.)

If B(x,y) = Sum_{n,k>=0} T(n,k)*x^n*y^k (with T(0,0) = 1, T(0,k) = 0 for k>=1, and T(n,k) = 0 for 1 <= n <= k), then B(x,y) = 1 + (x/2)*(B(x,y)^2/(1-x*y*B(x,y)) + (1 + x*y*B(x,y))*B(x^2,y^2)/(1-x^2*y^2*B(x^2,y^2))). This can be derived from the bivariate g.f. of A232206. See the comments for that sequence.

Let S(n) := Sum_{k>=0} T(n,k). The g.f. of S(n) is B(x, y=1). If we let y=1 in the above functional equation, we get x*B(x,1) = x + (1/2)*((x*B(x,1))^2/(1-x*B(x,1)) + (1 + x*B(x,1))*x^2*B(x^2,1)/(1-x^2*B(x^2,1))). After some algebra, we get 2*x*B(x,1) = x + (1/2)(x*B(x,1)/(1-x*B(x,1)) + (x*B(x,1) + x^2*B(x^2,1))/(1-x^2*B(x,1))), i.e., 2*x*B(x,1) = x + BIK(x*B(x,1)), where we have the "BIK" (reversible, indistinct, unlabeled) transform of C. G. Bower. This proves that S(n) = A032132(n+1) for n>=0, which is Copeland's claim above.

Note that for the second column we have T(n,k=2) = A048739(n-2) for 2 <= n < = 10, but T(11,2) = 4062 <> 4059 = A048739(9). In any case, they have different g.f.s (see the formula section below).

(End)