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

Molien series of 2-dimensional representation of cyclic group of order 3 over GF(2).

One step back, two steps forward.

The crossing number of the graph C(n, {1,3}), n >= 8, is [n/3] + n mod 3, which gives this sequence starting at the first 4. [Yang Yuansheng et al.]

A Chebyshev transform of A078008. The g.f. is the image of (1-x)/(1-x-2*x^2) (g.f. of A078008) under the Chebyshev transform A(x)-> (1/(1+x^2))*A(x/(1+x^2)). - Paul Barry, Oct 15 2004

A047878 is an essentially identical sequence. - Anton Chupin, Oct 24 2009

Rhyme scheme of Dante Alighieri's "Divine Comedy." - David Gaita, Feb 11 2011

A194960 results from deleting the first four terms of A008611. Note that deleting the first term or first four terms of A008611 leaves a concatenation of segments (n, n+1, n+2); for related concatenations, see

A008619, (n,n+1) after deletion of first term;

A053737, (n,n+1,n+2,n+3) beginning with n=0;

A053824, (n to n+4) beginning with n=0. - Clark Kimberling, Sep 07 2011

It appears that a(n) is the number of roots of x^(n+1) + x + 1 inside the unit circle. - Michel Lagneau, Nov 02 2012

Also apparently for n >= 2: a(n) is the largest remainder r that results from dividing n+2 by 1..n+2 more than once, i.e., a(n) = max(i, A072528(n+2,i)>1). - Ralf Stephan, Oct 21 2013

Number of n-element subsets of [n+1] whose sum is a multiple of 3. a(4) = 1: {1,2,4,5}. - Alois P. Heinz, Feb 06 2017

It appears that a(n) is the number of roots of the Fibonacci polynomial F(n+2,x) strictly inside the unit circle of the complex plane. - Michel Lagneau, Apr 07 2017

For the proof of the preceding conjecture see my comments under A008615 and A049310. Chebyshev S(n,x) = i^n*F(n+1,-i*x), with i = sqrt(-1). - Wolfdieter Lang, May 06 2017

The sequence is the interleaving of three sequences: the positive integers (A000027), the nonnegative integers (A001477), and the positive integers, in that order. - Guenther Schrack, Nov 07 2020

a(n) is the number of multiples of 3 between n and 2n. - Christian Barrientos, Dec 20 2021

a(n) is the least number of football games a team has to play to be able to get n-1 points, where a win is 3 points, a draw is 1 point, and a loss is 0 points. - Sigurd Kittilsen, Dec 01 2022

Alkane (or paraffin) numbers l(6,n).

Dimension of the space of homogeneous degree n polynomials in (x1, y1, x2, y2) invariant under permutation of variables x1<->y1, x2<->y2.

Also multidigraphs with loops on 2 nodes with n arcs (see A138107). - Vladeta Jovovic, Dec 27 1999

Euler transform of finite sequence [2,3,0,-1]. - Michael Somos, Mar 17 2004

a(n-2) is the number of plane partitions with trace 2. - Michael Somos, Mar 17 2004

With offset 4, a(n) is the number of bracelets with n beads, 3 of which are red, 1 of which is blue. For odd n, a(n) = C(n-1,3)/2. For even n, a(n) = C(n-1,3)/2 +(n-2)/4. For n >= 6, with K = (n-1)(n-2)/((n-5)(n-4)), for odd n, a(n) = K*a(n-2). For even n, a(n) = K*a(n-2) -(n-2)/(n-5). - Washington Bomfim, Aug 05 2008

Equals (1,2,3,4,...) convolved with (1,0,3,0,5,...). - Gary W. Adamson, Feb 16 2009

Equals row sums of triangle A177878.

Equals (1/2)*((1, 4, 10, 20, 35, 56, ...) + (1, 0, 2 0, 3, 0, 4, ...)).

From Ctibor O. Zizka, Nov 21 2014: (Start)

With offset 4, a(n) is the number of different patterns of the 2-color 4-partition of n.

P(n)_(k;t) gives the number of different patterns of the t-color, k-partition of n.

P(n)(k;t) = 1 + Sum(i=2..n) Sum(j=2..i) Sum(r=1..m) c(i,j)*v_r*F_r(X_1,...,X_i).

P(n;i;j) = Sum(r=1..m) c_(i,j)*v_r*F_r(X_1,...,X_i).

m partition number of i.

c_(i,j) number of different coloring patterns on the r-th form (X_1,...,X_i) of i-partition with j-colors.

v_r number of i-partitions of n of the r-th form (X_1,...,X_i).

F_r(X_1,...,X_i) number of different patterns of the r-th form i-partition of n.

Some simple results:

P(1)(k;t)=1, P(2)(k;t)=2, P(3)(k;t)=4, P(4)(k;t)=11, etc.

P(n;1;1) = P(n;n;n) = 1 for all n;

P(n;2;2) = floor(n/2) (A004526);

P(n;3;2) = (n*n - 2*n + n mod 2)/4 (A002620).

This sequence is a(n) = P(n;4;2).

2-coloring of 4-partition is (A,B,A,B) or (B,A,B,A).

Each 4-partition of n has one of the form (X_1,X_1,X_1,X_1),(X_1,X_1,X_1,X_2), (X_1,X_1,X_2,X_2),(X_1,X_1,X_2,X_3),(X_1,X_2,X_3,X_4).

The number of forms is m=5 which is the partition number of k=4.

Partition form (X_1,X_1,X_1,X_1) gives 1 pattern ((X_1A,X_1B,X_1A,X_1B), (X_1,X_1,X_1,X_2) gives 2 patterns, (X_1,X_1,X_2,X_2) gives 4 patterns, (X_1,X_1,X_2,X_3) gives 6 patterns and (X_1,X_2,X_3,X_4) gives 12 patterns.

Thus a(n) = P(n;4;2) = 1*1*v_1 + 1*2*v_2 + 1*4*v_3 + 1*6*v_4 + 1*12*v_5 where v_r is the number of different 4-partitions of the r-th form (X_1,X_2,X_3,X_4) for a given n.

Example:

The 4-partitions of 8 are (2,2,2,2), (1,1,1,5), (1,1,3,3), (1,1,2,4), and (1,2,2,3):

(2,2,2,2) 1 pattern

(1,1,1,5), (1,1,5,1) 2 patterns

(1,1,3,3), (1,3,3,1), (3,1,1,3), (1,3,1,3) 4 patterns

(1,1,2,4), (1,1,4,2), (1,2,1,4), (1,2,4,1), (1,4,1,2), (2,1,1,4) 6 patterns

(2,2,1,3), (2,2,3,1), (2,1,2,3), (2,1,3,2), (2,3,2,1), (1,2,2,3) 6 patterns

Thus a(8) = P(8,4,2) = 1 + 2 + 4 + 6 + 6 = 19. (End)

a(n) = length of run n+2 of consecutive 1's in A254338. - Reinhard Zumkeller, Feb 27 2015

Take a chessboard of (n+2) X (n+2) unit squares in which the a1 square is black. a(n) is the number of composite squares having black unit squares on their vertices. - Ivan N. Ianakiev, Jul 19 2018

a(n) is the number of 1423-avoiding odd Grassmannian permutations of size n+2. Avoiding any of the patterns 2314 or 3412 gives the same sequence. - Juan B. Gil, Mar 09 2023

a(n) is also the number of median-reflective (palindrome) symmetric patterns in a top-down equilateral triangular arrangement of closely packed black and white cells satisfying the local matching rule of Pascal's triangle modulo 2, where n is the number of cells in each edge of the arrangement. The matching rule is such that any elementary top-down triangle of three neighboring cells in the arrangement contains either one or three white cells.

The number of possibilities for an n-game (sub)set of tennis with neither player gaining a 2-game advantage. (Motivated by the marathon Isner-Mahut match at Wimbledon, 2010.) - Barry Cipra, Jun 28 2010

Number of achiral rows of n colors using up to two colors. For a(3)=4, the rows are AAA, ABA, BAB, and BBB. - Robert A. Russell, Nov 07 2018

Also the number of walks of length n on the graph x--y--z starting at y. - Sean A. Irvine, May 30 2025

First differences of the generalized heptagonal numbers A085787. - Omar E. Pol, Sep 10 2011

Last term in n-th row of A080511.

Also A005408 and positive terms of A008585 interleaved. - Omar E. Pol, May 28 2012

a(n) is also the length of the n-th line segment of the rectangular spiral whose vertices are the generalized heptagonal numbers. - Omar E. Pol, Jul 27 2018

Powers of 2 with additional first two terms.

Essentially the same as A131577 (and A000079).

[(-1)^n*a(n)] = [1, 0, 1, -2, 4, -8, 16, -32, ...] is the inverse binomial transform of A008619 = [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, ...]. - Philippe Deléham, Nov 15 2009

Number of compositions (ordered partitions) of n into an even number of parts. - Geoffrey Critzer, Mar 28 2010

Number of compositions of n into an even number of even parts.

Number of compositions of n into parts k >= 2 where there are k - 1 sorts of part k. - Joerg Arndt, Sep 30 2012

Taking n-th differences of this sequence reproduces the same sequence except for a(1) = n mod 2 (parity of n) and a(0) = (-1)^a(1)*floor(n/2 + 1). - M. F. Hasler, Jan 13 2015

From Emeric Deutsch, Dec 14 2008: (Start)

Number of permutations of {1,2,...,n-1} having a single run of odd entries. Example: a(5)=12 because we have 1324,1342,3124,3142,2134,4132,2314,4312, 2413, 4213, 2431 and 4231.

a(n) = A152666(n-1,1). (End)

a(n+1) gives the permanent of the n X n matrix whose (i,j)-element is i+j-1 modulo 2. - John W. Layman, Jan 03 2011

From Daniel Forgues, May 20 2011: (Start)

a(0) = 1 since it is the empty product.

A010551(n-2), n >= 2, equal to (ceiling((n-2)/2))! * (floor((n-2)/2))!, gives the number of arrangements of n-2 entries from 2 to n-1, starting with an even entry and where the parity of adjacent entries alternates. This is the number of arrangements to investigate for row n of a prime pyramid (A051237). (End)

Partial products of A008619. - Reinhard Zumkeller, Apr 02 2012

Also size of the equivalence class of S_n containing the identity permutation under transformations of positionally adjacent elements of the form abc <--> acb where a < b < c, cf. A210667 (equivalently under such transformations of the form abc <--> bac where a < b < c.) - Tom Roby, May 15 2012

Row sums of A246117. - Peter Bala, Aug 15 2014

a(n) is the number of parity-alternating permutations of size n. A permutation is parity-alternating if it sends even integers to even, and odd to odd. - Per W. Alexandersson, Jun 06 2022

n divides a(n) if and only if n is not prime. Since a(n) = floor(n/2)!*floor((n+1)/2)!, if n is prime then n is not a factor of a(n). All the prime factors of a(n) are in fact less than or equal to (n+1)/2. If n is composite, then it's possible to write it as p*q with p and q less than or equal to n/2. So p and q are factors of a(n). - Davide Oliveri, Apr 01 2023

Number of permutations of {1, 2, ..., n-1} where each entry is not greater than twice the previous entry. - Dewangga Putra Sheradhien, Jul 13 2024

a(n) is the number of nonnegative integer solutions to the equation x+y+z=n such that x+y=z. - Geoffrey Critzer, Jul 12 2013

For n >= i, i = 6,7, a(n - i) is the number of incongruent two-color bracelets of n beads, i of which are black (cf. A005513, A032280), having a diameter of symmetry. The latter means the following: if we imagine (0,1)-beads as points (with the corresponding labels) dividing a circumference of a bracelet into n identical parts, then a diameter of symmetry is a diameter (connecting two beads or not) such that a 180-degree turn of one of two sets of points around it (obtained by splitting the circumference by this diameter) leads to the coincidence of the two sets (including their labels). - Vladimir Shevelev, May 03 2011

From Johannes W. Meijer, May 20 2011: (Start)

The Kn11, Kn12, Kn13, Fi1 and Ze1 triangle sums, see A180662 for their definitions, of the Connell-Pol triangle A159797 are linear sums of shifted versions of the duplicated tetrahedral numbers, e.g., Fi1(n) = a(n-1) + 5*a(n-2) + a(n-3) + 5*a(n-4).

The Kn11, Kn12, Kn13, Kn21, Kn22, Kn23, Fi1, Fi2, Ze1 and Ze2 triangle sums of the Connell sequence A001614 as a triangle are also linear sums of shifted versions of the sequence given above. (End)

The number of quadruples of integers [x, u, v, w] that satisfy x > u > v > w >= 0, n + 5 = x + u. - Michael Somos, Feb 09 2015

Also, this sequence is the fourth column in the triangle of the coefficients of the sum of two consecutive Fibonacci polynomials F(n+1, x) and F(n, x) (n>=0) in ascending powers of x. - Mohammad K. Azarian, Jul 18 2018

Number of odd parts in all compositions (ordered partitions) of n: a(3)=6 because in 3=2+1=1+2=1+1+1 we have 6 odd parts. Number of even parts in all compositions (ordered partitions) of n+1: a(3)=6 because in 4=3+1=1+3=2+2=2+1+1=1+2+1=1+1+2=1+1+1+1 we have 6 even parts.

Convolved with (1, 2, 2, 2, ...) = A001787: (1, 4, 12, 32, 80, ...). - Gary W. Adamson, May 23 2009

An elephant sequence, see A175654. For the corner squares 36 A[5] vectors, with decimal values between 15 and 480, lead to this sequence. For the central square these vectors lead to the companion sequence 4*A172481, for n>=-1. - Johannes W. Meijer, Aug 15 2010

a(n) is the total number of runs of equal parts in the compositions of n. a(5) = 34 because there are 34 runs of equal parts in the compositions of 5, with parentheses enclosing each run: (5), (4)(1), (1)(4), (3)(2), (2)(3), (3)(1,1), (1)(3)(1), (1,1)(3), (2,2)(1), (2)(1)(2), (1)(2,2), (2)(1,1,1), (1)(2)(1,1), (1,1)(2)(1), (1,1,1)(2), (1,1,1,1,1). - Gregory L. Simay, Apr 28 2017

a(n) - a(n-2) is the number of 1's in all compositions of n and more generally, the number of k's in all compositions of n+k-1. - Gregory L. Simay, May 01 2017