A007581 a(n) = (2^n+1)*(2^n+2)/6. (Formerly M1479)

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

0,2

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).

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

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 *)

Prog

(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
(Magma) [(2^n+1)*(2^n+2)/6: n in [0..25]]; // Vincenzo Librandi, Aug 09 2018
(GAP) List([0..30], n->(2^n+1)*(2^n+2)/6); # Muniru A Asiru, Aug 09 2018

Crossrefs

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

A row of the array in A278984.

Sequence in context: A149953 A149954 A149955 * A124303 A367414 A073525

Adjacent sequences: A007578 A007579 A007580 * A007582 A007583 A007584

Keyword

nonn,easy,nice

Author

Simon Plouffe and N. J. A. Sloane

Status

approved