A007581 a(n) = (2^n+1)*(2^n+2)/6.

1, 2, 5, 15, 51, 187, 715, 2795, 11051, 43947, 175275, 700075, 2798251, 11188907, 44747435, 178973355, 715860651, 2863377067, 11453377195, 45813246635, 183252462251, 733008800427, 2932033104555, 11728128223915, 46912504507051, 187650001250987

Offset: 0

Author

Simon Plouffe and N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

Number of palindromic structures using a maximum of four different symbols. - Marks R. Nester

Dimension of the universal embedding of the symplectic dual polar space DSp(2n,2) (conjectured by A. Brouwer, proved by P. Li). - J. Taylor (jt_cpp(AT)yahoo.com), Apr 02 2004.

Apart from initial term, same as A124303. - Valery A. Liskovets, Nov 16 2006

Hankel transform is := [1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...]. - Philippe Deléham, Dec 04 2008

a(n) is also the number of distinct solutions (avoiding permutations) to the equation: XOR(A,B,C)=0 where A,B,C are n-bit binary numbers. - Ramasamy Chandramouli, Jan 11 2009

The rank of the fundamental group of the Z_p^n - cobordism category in dimension 1+1 for the case p=2 (see paper below). The expression for any prime p is (p^(2n-1)+p^(n+1)-p^(n-1)+p^2-p-1)/(p^2-1). - Carlos Segovia Gonzalez, Dec 05 2012

The number of isomorphic classes of regular four coverings of a graph with respect to the identity automorphism (S. Hong and J. H. Kwak). - Carlos Segovia Gonzalez, Aug 01 2013

The density of a language with four letters (N. Moreira and R. Reis). - Carlos Segovia Gonzalez, Aug 01 2013

References

• P. Li, On the Brouwer Conjecture for Dual Polar Spaces of Symplectic Type over GF(2). Preprint.
• M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 0..200
• Joerg Arndt and N. J. A. Sloane, Counting Words that are in "Standard Order"
• Barnes, Jeffrey M.; Benkart, Georgia; Halverson, Tom McKay centralizer algebras. Proc. Lond. Math. Soc. (3) 112, No. 2, 375-414 (2016).
• Georgia Benkart and Tom Halverson, McKay Centralizer Algebras, hal-02173744 [math.CO], 2020.
• A. Blokhuis and A. E. Brouwer, The universal embedding dimension of the binary symplectic dual polar space, Discr. Math., 264 (2003), 3-11.
• P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
• Bruno Cisneros, Carlos Segovia, An approximation for the number of subgroups, arXiv:1805.04633 [math.GT], 2018.
• B. N. Cooperstein and E. E. Shult, A note on embedding and generating dual polar spaces. Adv. Geom. 1 (2001), 37-48.
• Yuanan Diao, Michael Finney, and Dawn Ray, The number of oriented rational links with a given deficiency number, arXiv:2007.02819 [math.GT], 2020. See p. 16.
• A. M. Hinz, S. Klavžar, U. Milutinović, and C. Petr, The Tower of Hanoi - Myths and Maths, Birkhäuser 2013. See page 183. Book's website
• S. Hong and J. H. Kwak, Regular fourfold covering with respect to the identity automorphism, J. Graph Theory, 17 (1993), 621-627.
• Masashi Kosuda and Manabu Oura, Centralizer algebras of the primitive unitary reflection group of order 96, arXiv:1505.00318 [math.RT], 2015.
• George S. Lueker, Improved Bounds on the Average Length of Longest Common Subsequences (Jul 22, 2005) (Fig.1).
• N. Moreira and R. Reis, On the Density of Languages Representing Finite Set Partitions, Journal of Integer Sequences, Vol. 8 (2005), Article 05.2.8.
• C. Segovia, Unexpected relations of cobordism categories with another [sic] subjects in mathematics, 2013.
• C. Segovia, The classifying space of the 1+1 dimensional G-cobordism category, arXiv:1211.2144 [math.AT], 2012-2019.
• C. Segovia, Numerical computations in cobordism categories, arXiv:1307.2850 [math.AT], 2013.
• C. Segovia and M. Winklmeier, Combinatorial Computations in Cobordism Categories, arXiv preprint arXiv:1409.2067 [math.CO], 2014-2015.
• C. Segovia and M. Winklmeier, On the density of certain languages with p^2 letters, Electronic Journal of Combinatorics 22(3) (2015), #P3.16.
• Index entries for linear recurrences with constant coefficients, signature (7,-14,8).

Crossrefs

Cf. A056272, A056273, A007051, A000392, A056450, A028401, A060919.

A row of the array in A278984.

Programs

GAP

List([0..30],n->(2^n+1)*(2^n+2)/6); # Muniru A Asiru, Aug 09 2018

Magma

[(2^n+1)*(2^n+2)/6: n in [0..25]]; // Vincenzo Librandi, Aug 09 2018

Maple

A007581:=n->(2^n+1)*(2^n+2)/6; seq(A007581(n), n=0..50); # Wesley Ivan Hurt, Nov 25 2013

Mathematica

Table[(3*2^(n-1)+2^(2n-1)+1)/3,{n,0,30}] (* or *) LinearRecurrence[ {7,-14,8},{1,2,5},31] (* Harvey P. Dale, Jul 24 2011 *)
CoefficientList[Series[(1 - 5 x + 5 x^2) / ((1 - x) (1 - 2 x) (1 - 4 x)), {x, 0, 33}], x] (* Vincenzo Librandi, Aug 09 2018 *)

PARI

a(n)=(3*2^(n-1)+2^(2*n-1)+1)/3; \\ Charles R Greathouse IV, Jun 24 2011

PARI

a(n)=if(n==0,1,(2<<(2*n--))\/3+2^n); \\ Charles R Greathouse IV, Jun 24 2011

Formula

a(n) = (3*2^(n-1) + 2^(2*n-1) + 1)/3.

a(n) = Sum_{k=1..4} Stirling2(n, k). - Winston Yang (winston(AT)cs.wisc.edu), Aug 23 2000

Binomial transform of 3^n/6 + 1/2 + 0^n/3, i.e., of A007051 with an extra leading 1. a(n) = binomial(2^n+2, 2^n-1)/2^n. - Paul Barry, Jul 19 2003

a(n) = C(2+2^n, 3)/2^n = a(n-1) + 2^(n-1) + 4^(n-3/2) = A092055(n)/A000079(n). - Henry Bottomley, Feb 19 2004

Second binomial transform of A001045(n-1) + 0^n/2. G.f.: (1-5*x+5*x^2)/((1-x)*(1-2*x)*(1-4*x)). - Paul Barry, Apr 28 2004

a(n) is the top entry of the vector M^n*[1,1,1,1,0,0,0,...], where M is an infinite bidiagonal matrix with M(r,r)=r, r >= 1, as the main diagonal, M(r,r+1)=1, and the rest zeros. ([1,1,1,...] is a column vector and transposing gives the same in terms of a leftmost column term.) - Gary W. Adamson, Jun 24 2011

a(0)=1, a(1)=2, a(2)=5, a(n) = 7*a(n-1) - 14*a(n-2) + 8*a(n-3). - Harvey P. Dale, Jul 24 2011

E.g.f.: (exp(2*x) + 1/3*exp(4*x) + 2/3*exp(x))/2 = G(0)/2; G(k)=1 + (2^k)/(3 - 6/(2 + 4^k - 3*x*(8^k)/(3*x*(2^k) + (k+1)/G(k+1)))); (continued fraction). - Sergei N. Gladkovskii, Dec 08 2011