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%I M1479 #153 Sep 08 2022 08:44:35
%S 1,2,5,15,51,187,715,2795,11051,43947,175275,700075,2798251,11188907,
%T 44747435,178973355,715860651,2863377067,11453377195,45813246635,
%U 183252462251,733008800427,2932033104555,11728128223915,46912504507051,187650001250987
%N a(n) = (2^n+1)*(2^n+2)/6.
%C Number of palindromic structures using a maximum of four different symbols. - _Marks R. Nester_
%C 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.
%C Apart from initial term, same as A124303. - _Valery A. Liskovets_, Nov 16 2006
%C Hankel transform is := [1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...]. - _Philippe Deléham_, Dec 04 2008
%C 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
%C 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
%C 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
%C The density of a language with four letters (N. Moreira and R. Reis). - _Carlos Segovia Gonzalez_, Aug 01 2013
%D P. Li, On the Brouwer Conjecture for Dual Polar Spaces of Symplectic Type over GF(2). Preprint.
%D 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]
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A007581/b007581.txt">Table of n, a(n) for n = 0..200</a>
%H Joerg Arndt and N. J. A. Sloane, <a href="/A278984/a278984.txt">Counting Words that are in "Standard Order"</a>
%H Barnes, Jeffrey M.; Benkart, Georgia; Halverson, Tom <a href="https://doi.org/10.1112/plms/pdv075">McKay centralizer algebras</a>. Proc. Lond. Math. Soc. (3) 112, No. 2, 375-414 (2016).
%H Georgia Benkart and Tom Halverson, <a href="https://hal.archives-ouvertes.fr/hal-02173744">McKay Centralizer Algebras</a>, hal-02173744 [math.CO], 2020.
%H A. Blokhuis and A. E. Brouwer, <a href="http://dx.doi.org/10.1016/S0012-365X(02)00545-9">The universal embedding dimension of the binary symplectic dual polar space</a>, Discr. Math., 264 (2003), 3-11.
%H P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/groups.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H Bruno Cisneros, Carlos Segovia, <a href="https://arxiv.org/abs/1805.04633">An approximation for the number of subgroups</a>, arXiv:1805.04633 [math.GT], 2018.
%H B. N. Cooperstein and E. E. Shult, <a href="http://www.emis.de/journals/AG/1-1/1_037.pdf">A note on embedding and generating dual polar spaces</a>. Adv. Geom. 1 (2001), 37-48.
%H Yuanan Diao, Michael Finney, and Dawn Ray, <a href="https://arxiv.org/abs/2007.02819">The number of oriented rational links with a given deficiency number</a>, arXiv:2007.02819 [math.GT], 2020. See p. 16.
%H A. M. Hinz, S. Klavžar, U. Milutinović, and C. Petr, <a href="http://dx.doi.org/10.1007/978-3-0348-0237-6">The Tower of Hanoi - Myths and Maths</a>, Birkhäuser 2013. See page 183. <a href="http://tohbook.info">Book's website</a>
%H S. Hong and J. H. Kwak, <a href="http://dx.doi.org/10.1002/jgt.3190170509">Regular fourfold covering with respect to the identity automorphism</a>, J. Graph Theory, 17 (1993), 621-627.
%H Masashi Kosuda and Manabu Oura, <a href="http://arxiv.org/abs/1505.00318">Centralizer algebras of the primitive unitary reflection group of order 96</a>, arXiv:1505.00318 [math.RT], 2015.
%H George S. Lueker, <a href="http://www.ics.uci.edu/~lueker/papers/lcs/lcs.pdf">Improved Bounds on the Average Length of Longest Common Subsequences</a> (Jul 22, 2005) (Fig.1).
%H N. Moreira and R. Reis, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL8/Moreira/moreira8.html">On the Density of Languages Representing Finite Set Partitions</a>, Journal of Integer Sequences, Vol. 8 (2005), Article 05.2.8.
%H C. Segovia, <a href="https://www.matem.unam.mx/~csegovia/Arch/Poster.pdf">Unexpected relations of cobordism categories with another [sic] subjects in mathematics</a>, 2013.
%H C. Segovia, <a href="http://arxiv.org/abs/1211.2144">The classifying space of the 1+1 dimensional G-cobordism category</a>, arXiv:1211.2144 [math.AT], 2012-2019.
%H C. Segovia, <a href="http://arxiv.org/abs/1307.2850">Numerical computations in cobordism categories</a>, arXiv:1307.2850 [math.AT], 2013.
%H C. Segovia and M. Winklmeier, <a href="http://arxiv.org/abs/1409.2067">Combinatorial Computations in Cobordism Categories</a>, arXiv preprint arXiv:1409.2067 [math.CO], 2014-2015.
%H C. Segovia and M. Winklmeier, <a href="https://doi.org/10.37236/4668">On the density of certain languages with p^2 letters</a>, Electronic Journal of Combinatorics 22(3) (2015), #P3.16.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (7,-14,8).
%F a(n) = (3*2^(n-1) + 2^(2*n-1) + 1)/3.
%F a(n) = Sum_{k=1..4} Stirling2(n, k). - Winston Yang (winston(AT)cs.wisc.edu), Aug 23 2000
%F 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
%F 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
%F 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
%F 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
%F 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
%F 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
%p A007581:=n->(2^n+1)*(2^n+2)/6; seq(A007581(n), n=0..50); # _Wesley Ivan Hurt_, Nov 25 2013
%t 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 *)
%t 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 *)
%o (PARI) a(n)=(3*2^(n-1)+2^(2*n-1)+1)/3; \\ _Charles R Greathouse IV_, Jun 24 2011
%o (PARI) a(n)=if(n==0,1,(2<<(2*n--))\/3+2^n); \\ _Charles R Greathouse IV_, Jun 24 2011
%o (Magma) [(2^n+1)*(2^n+2)/6: n in [0..25]]; // _Vincenzo Librandi_, Aug 09 2018
%o (GAP) List([0..30],n->(2^n+1)*(2^n+2)/6); # _Muniru A Asiru_, Aug 09 2018
%Y Cf. A056272, A056273, A007051, A000392, A056450, A028401, A060919.
%Y A row of the array in A278984.
%K nonn,easy,nice
%O 0,2
%A _Simon Plouffe_ and _N. J. A. Sloane_
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