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A007581
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(3*2^(n-1) + 2^(2*n-1) + 1)/3.
(Formerly M1479)
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23
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1, 2, 5, 15, 51, 187, 715, 2795, 11051, 43947, 175275, 700075, 2798251, 11188907, 44747435, 178973355, 715860651, 2863377067, 11453377195, 45813246635, 183252462251, 733008800427, 2932033104555, 11728128223915
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Number of palindromic structures using a maximum of four different symbols. - Marks R. Nester (nesterm(AT)dpi.qld.gov.au)
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 (liskov(AT)im.bas-net.by), Nov 16 2006
Hankel transform is := [1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...]. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 04 2008]
Contribution from Ramasamy Chandramouli (thedavinci(AT)gmail.com), Jan 11 2009: (Start)
a(n) is also the number of unique solutions (avoiding permutations) to the equation: XOR(A,B,C)=0 where A,B,C are n-bit binary numbers.
(End)
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REFERENCES
| A. Blokhuis and A. E. Brouwer, The universal embedding dimension of the binary symplectic dual polar space, Discr. Math., 264 (2003), 3-11.
S. Hong and J. H. Kwak, Regular fourfold covering with respect to the identity automorphism, J. Graph Theory, 17 (1993), 621-627.
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.
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.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..200
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
B. N. Cooperstein and E. E. Shult, A note on embedding and generating dual polar spaces. Adv. Geom. 1 (2001), 37-48.
George S. Lueker, Improved Bounds on the Average Length of Longest Common Subsequences (Jul 22, 2005) (Fig.1).
Index to sequences with linear recurrences with constant coefficients, signature (7,-14,8).
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FORMULA
| (2^n+1)*(2^n+2)/6.
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 (pbarry(AT)wit.ie), 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 (se16(AT)btinternet.com), 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 (pbarry(AT)wit.ie), 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) [From 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
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MATHEMATICA
| Table[(3*2^(n-1)+2^(2n-1)+1)/3, {n, 0, 30}] (* or *) LinearRecurrence[ {7, -14, 8}, {1, 2, 5}, 31] (* From Harvey P. Dale, Jul 24 2011 *)
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PROG
| (PARI) a(n)=(3*2^(n-1)+2^(2*n-1)+1)/3 \\ Charles R Greathouse IV, Jun 24 2011
(PARI) a(n)=(2<<(2*n--))\/3+2^n \\ for n > 0, Charles R Greathouse IV, Jun 24 2011
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CROSSREFS
| Cf. A056272, A056273, A007051, A000392, A056450, A028401, A060919.
Sequence in context: A149953 A149954 A149955 * A124303 A073525 A007317
Adjacent sequences: A007578 A007579 A007580 * A007582 A007583 A007584
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KEYWORD
| nonn,easy,nice
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AUTHOR
| Simon Plouffe, N. J. A. Sloane (njas(AT)research.att.com).
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