cp's OEIS Frontend

Coefficient of x^(2n-2) in Chebyshev polynomial T(2n) is -a(n).

Let M_n be the n X n matrix m_(i,j) = 1 + 2*abs(i-j); then det(M_n) = (-1)^(n-1)*a(n-1). - Benoit Cloitre, May 28 2002

Number of subsequences 00 in all words of length n+1 on the alphabet {0,1,2,3}. Example: a(2)=8 because we have 000,001,002,003,100,200,300 (the other 57=A125145(3) words of length 3 have no subsequences 00). a(n) = Sum_{k=0..n} k*A128235(n+1, k). - Emeric Deutsch, Feb 27 2007

Let P(A) be the power set of an n-element set A. Then a(n) = the sum of the size of the symmetric difference of x and y for every subset {x,y} of P(A). - Ross La Haye, Dec 30 2007 (See the comment from Bernard Schott below.)

Let P(A) be the power set of an n-element set A and B be the Cartesian product of P(A) with itself. Then remove (y,x) from B when (x,y) is in B and x != y and call this R35. Then a(n) = the sum of the size of the symmetric difference of x and y for every (x,y) of R35. [proposed edit of comment just above; by Ross La Haye]

The numbers in this sequence are the Wiener indices of the graphs of n-cubes (Boolean hypercubes). For example, the 3-cube is the graph of the standard cube whose Wiener index is 48. - K.V.Iyer, Feb 26 2009

From Gary W. Adamson, Sep 06 2009: (Start)

Starting (1, 8, 48, ...) = 4th binomial transform of [1, 4, 0, 0, 0, ...].

Equals the sum of terms in 2^n X 2^n semi-magic square arrays in which each row and column is composed of a binomial frequency of terms in the set (1, 3, 5, 7, ...).

The first few such arrays = [1] [1,3; 3,1]; /Q.

[1, 3, 5, 3;

3, 1, 3, 5;

5, 3, 1, 3;

3, 5, 3, 1]

(sum of terms = 48, with a binomial frequency of (1, 2, 1) as to (1, 3, 5) in each row and column)

[1, 3, 5, 3, 5, 7, 5, 3;

3, 1, 3, 5, 7, 5, 3, 5;

5, 3, 1, 3, 5, 3, 5, 7;

3, 5, 3, 1, 3, 5, 7, 5;

5, 7, 5, 3, 1, 3, 5, 3;

7, 5, 3, 5, 3, 1, 3, 5;

5, 3, 5, 7, 5, 3, 1, 3;

3, 5, 7, 5, 3, 5, 3, 1]

(sum of terms = 256, with each row and column composed of one 1, three 3's, three 5's, and one 7)

... (End)

Let P(A) be the power set of an n-element set A and B be the Cartesian product of P(A) with itself. Then a(n) = the sum of the size of the intersection of x and y for every (x,y) of B. - Ross La Haye, Jan 05 2013

Following the last comment of Ross, A002699 is the similar sequence when "intersection" is replaced by "symmetric difference" and A212698 is the similar sequence when "intersection" is replaced by "union". - Bernard Schott, Jan 04 2013

Also, following the first comment of Ross, A082134 is the similar sequence when "symmetric difference" is replaced by "intersection" and A133224 is the similar sequence when "symmetric difference" is replaced by "union". - Bernard Schott, Jan 15 2013

Let [n] denote the set {1,2,3,...,n} and denote an n-permutation of the elements of [n] by p = p(1)p(2)p(3)...p(n), where p(i) is the i-th entry in the linear order given by p. Then (p(i),p(j)) is an inversion of p if i < j but p(i) > p(j). Denote the number of inversions of p by inv(p) and call a 2n-permutation p = p(1)p(2)...p(2n) 2-ordered if p(1) < p(3) < ... < p(2n-1) and p(2) < p(4) < ... < p(2n). Then Sum(inv(p)) = n*4^(n-1), where the sum is taken over all 2-ordered 2n-permutations of p. See Bona reference below. - Ross La Haye, Jan 21 2014

Sum over all peaks of Dyck paths of semilength n of the product of the x and y coordinates. - Alois P. Heinz, May 29 2015

Sum of the number of all edges over all j-dimensional subcubes of the boolean hypercube graph of dimension n, Q_n, for all j, so a(n) = Sum_{j=1..n} binomial(n,j)*2^(n-j) * j*2^(j-1). - Constantinos Kourouzides, Mar 24 2024

Number of vertical pairs in a wheel with n equal sections. - Wesley Ivan Hurt, Jan 22 2012

Number of even terms of n-th row in the triangles A162610 and A209297. - Reinhard Zumkeller, Jan 19 2013

Also the result of writing n-1 in base 2 and multiplying the last digit with the number with its last digit removed. See A115273 and A257844-A257850 for generalization to other bases. - M. F. Hasler, May 10 2015

Also follows the rule: a(n+1) is the number of terms that are identical with a(n) for a(0..n-1). - Marc Morgenegg, Jul 08 2019

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

Multiplicative with a(2^e)=0 if e>0 and a(p^e)=p^e for odd primes p. - R. J. Mathar, Aug 01 2011

A005408 and A000004 interleaved (the usual OEIS policy is not to include sequences like this where alternate terms are zero; this is an exception). - Omar E. Pol, Feb 02 2013

Row sums of A057211. - Omar E. Pol, Mar 05 2014

Column k=2 of triangle A196020. - Omar E. Pol, Aug 07 2015

a(n) is the determinant of the (n+2) X (n+2) circulant matrix with the first row [0,0,1,1,...,1]. This matrix is closely linked with the famous ménage problem (see also comments of Vladimir Shevelev in sequence A000179). Namely it defines the class of permutations p of 1,2,...,n+2 such that p(i)<>i and p(i)<>i+1 for i=1,2,...,n+1, and p(n+2)<>1,n+2. And a(n) is also the difference between the number of even and odd such permutations. - Dmitry Efimov, Feb 02 2016

Let M_n be the n X n matrix m_(i,j) = 3 + abs(i-j), then det(M_n) =(-1)^(n+1)*a(n+1). - Benoit Cloitre, May 28 2002

If X_1, X_2, ..., X_n are 2-blocks of a (2n+3)-set X then, for n>=1, a(n+2) is the number of (n+1)-subsets of X intersecting each X_i, (i=1..n). - Milan Janjic, Nov 18 2007

Equals row sums of triangle A152194. - Gary W. Adamson, Nov 28 2008

An elephant sequence, see A175655. For the central square 16 A[5] vectors, with decimal values between 19 and 400, lead to this sequence (without the first leading 1). For the corner squares these vectors lead to the companion sequence A045623. - Johannes W. Meijer, Aug 15 2010

a(n) is the total number of runs of 1 in the compositions of n+1. For example, a(3) = A045623(3) - A045623(2) = 12 - 5 = 7 runs of only 1 in the compositions of 4, enumerated "()" as follows: 3,(1); (1),3; 2,(1,1);(1),2,(1); (1,1),2; (1,1,1,1). More generally, the total number of runs of only part k in the compositions of n+k is A045623(n) - A045623(n-k). - Gregory L. Simay, May 02 2017

This is essentially the p-INVERT of (1,1,1,1,1,...) for p(S) = 1 - S - S^2 + S^3; see A291000. - Clark Kimberling, Aug 24 2017

The number of positive terms in row n is A001227(n).

If n = 2^j then the only positive integer in row n is T(n,1) = n

If n is an odd prime then the only two positive integers in row n are T(n,1) = n and T(n,2) = (n - 1)/2.

From Omar E. Pol, Apr 30 2017: (Start)

Conjecture 1: T(n,k) is the smallest part of the partition of n into k consecutive parts, if T(n,k) > 0.

Conjecture 2: the last positive integer in the row n is in the column A109814(n). (End)

The atomic numbers of the augmented alkaline earth group in Charles Janet's spiral periodic table are 0 and the first eight terms of this sequence (see Stewart reference). - Alonso del Arte, May 13 2011

Maximum number of 123 patterns in an alternating permutation of length n+3. - Lara Pudwell, Jun 09 2019

a(n) = b(n)/c(n) where b(n) = A001405(n+1) - A001405(n), c(n) = gcd(A001405(n+1), A001405(n)).

Also the minimal number of disjoint edge-paths into which the complete graph on n edges can be partitioned - Felix Goldberg (felixg(AT)tx.technion.ac.il), Jan 19 2001

For n >= 2, number of partitions of n-2 into parts that are distinct mod 2. - Giovanni Resta, Feb 06 2006

Sequence starting with a(3) obeys the rule "smallest positive value such that the ordered pair (a(n-1),a(n)) has not occurred previously", or the rule "smallest positive value such that the ratio a(n-1)/a(n) has not occurred previously". The same subsequence has its ordinal transform equal to itself, shifted left. (The ordinal transform has as its n-th term the number of values in a(1),...,a(n) that are equal to a(n).) - Franklin T. Adams-Watters, Dec 13 2006

Numerators of floor(n/2)/n, n > 0. - Wesley Ivan Hurt, Jun 14 2013

Number of nonisomorphic outer planar graphs of order n >= 3, maximum degree 3, and largest possible size. The size is (3n-2)/2 when n is even and (3n-3)/2 when n is odd. - Christian Barrientos and Sarah Minion, Feb 27 2018

Graded dimension of L''/[L',L''] for the free Lie algebra on 2 generators. Let L be a free Lie algebra with 2 generators graded by the total degree. Set L'=[L,L] and L''=[L',L']. Then a(n) is equal to the dimension of the homogeneous subspace of degree n+2 in the quotient L''/[L',L'']. - Sergei Duzhin, Mar 15 2004

Also the 2nd Witt transform of A000027. - R. J. Mathar, Nov 08 2008

Also the number of 3-element subsets of {1..n+1} whose elements sum up to an odd integer, i.e., the third column of A159916: e.g. a(3)=2 corresponds to the two subsets {1,2,4} and {2,3,4} of {1..4}. - M. F. Hasler, May 01 2009

The set of magic numbers for an idealized harmonic oscillator nucleus with a biaxially deformed prolate ellipsoid shape and an oscillator ratio of 2:1. - Jess Tauber, May 13 2013

Quasipolynomial of order 2. - Charles R Greathouse IV, May 14 2013

The terms for n>1 can also be defined by: a(n)=0 if n==0 (mod 3), and otherwise a(n) equals the inverse of 3 in Z/nZ*. - José María Grau Ribas, Jun 18 2013

The subsequence of nonzero terms is essentially the same as A026741. - Giovanni Resta, Jun 18 2013