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 A063376 a(-1) = 1; for n >= 0, a(n) = 2^n + 4^n = 2^n*(1 + 2^n). 59
 1, 2, 6, 20, 72, 272, 1056, 4160, 16512, 65792, 262656, 1049600, 4196352, 16781312, 67117056, 268451840, 1073774592, 4295032832, 17180000256, 68719738880, 274878431232, 1099512676352, 4398048608256, 17592190238720, 70368752566272 (list; graph; refs; listen; history; text; internal format)
 OFFSET -1,2 COMMENTS Counts closed walks of length 2n+2 at a vertex of the cyclic graph on 8 nodes C_8. The count of closed walks of odd length is zero. See the array w(N,L) and triangle a(K,N) given in A199571 for the general case. - Wolfdieter Lang, Nov 08 2011 Number of monic irreducible polynomials of degree 1 in GF(2^n)[x,y]. - Max Alekseyev, Jan 23 2006 a(n) written in base 2: a(-1) = 1, a(0) = 10, a(n) for n >= 1: 110, 10100, 1001000, 100010000, ..., i.e., number 1, (n-1) times 0, number 1, n times 0 (see A163664). a(n) for n >= 0 is duplicate of A161168. a(n) for n >= 0 is a bisection of A005418. - Jaroslav Krizek, Aug 14 2009 With offset 0 = binomial transform of A122983. - Gary W. Adamson, Apr 18 2011 REFERENCES B. N. Cyvin et al., Isomer enumeration of unbranched catacondensed polygonal systems with pentagons and heptagons, Match, No. 34 (Oct 1996), pp. 109-121. See Table 4. LINKS Harry J. Smith, Table of n, a(n) for n=-1,200 Georgia Benkart, Dongho Moon, Walks on Graphs and Their Connections with Tensor Invariants and Centralizer Algebras, arXiv preprint arXiv:1610.07837 [math.RT], 2016-2017. J. Brunvoll, S. J. Cyvin and B. N. Cyvin, Isomer enumeration of some polygonal systems representing polycyclic conjugated hydrocarbons, J. Molec. Struct. (Theochem), 364 (1996), 1-13. (See Table 11.) S. Capparelli, A. Del Fra, Dyck Paths, Motzkin Paths, and the Binomial Transform, Journal of Integer Sequences, 18 (2015), #15.8.5. B. N. Cyvin et al., Isomer enumeration of unbranched catacondensed polygonal systems with pentagons and heptagons, 1996 [Annotated scanned copy of pages 118, 119 only]. T. A. Gulliver, Sums of Powers of Integers Divisible by Three, Int. J. Contemp. Math. Sciences, Vol. 7, 2012, no. 38, 1895 - 1901. D. Suprijanto and Rusliansyah, Observation on Sums of Powers of Integers Divisible by Four, Applied Mathematical Sciences, Vol. 8, 2014, no. 45, 2219 - 2226. Index entries for linear recurrences with constant coefficients, signature (6,-8). FORMULA a(n) = Sum_{k=0..n} if((n-k) mod 4 = 0, binomial(n, 2k), 0)}. - Paul Barry, Sep 19 2005 a(n) = 4a(n-1) - 2^n = 6a(n-1) - 8a(n-2) = A001576(n) - 1 = 2*A007582(n) = A005418(2n+2) = A002378(A000079(n)). G.f.: 1/x + (2-6*x)/((1-2*x)*(1-4*x)). a(n) = ceiling(2^n*(2^n + 1)), n >= -1. - Zerinvary Lajos, Jan 07 2008 E.g.f.: exp(2*x)*cosh(x)^2. - Paul D. Hanna, Oct 25 2012 E.g.f.: (1+Q(0))/4, where Q(k) = 1 + 2/( 2^k - 2*x*4^k/( 2*x*2^k + (k+1)/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Dec 16 2013 EXAMPLE a(1)=6 counts six round trips from, say, vertex no 1: 12121, 18181, 12181, 18121, 12321 and 18781. - Wolfdieter Lang, Nov 08 2011 MAPLE seq(ceil(2^n*(2^n + 1)), n=-1..23); # Zerinvary Lajos, Jan 07 2008 MATHEMATICA Table[2^n + 4^n, {n, 0, 25}] PROG (PARI) a(n)={if(n>=0, 2^n*(1 + 2^n), 1)} \\ Harry J. Smith, Aug 20 2009 (PARI) {a(n)=n!*polcoeff((1 + exp(2*x +x*O(x^n)))^2/4, n)} \\ Paul D. Hanna, Oct 25 2012 CROSSREFS Cf. A000051, A006516, A007582, A034472, A034474, A034491, A052539, A062394, A062395, A062396, A007689, A063376, A063481, A074600 - A074624, A122983. A column of A323850. Sequence in context: A192658 A049141 A049129 * A161168 A049139 A071356 Adjacent sequences:  A063373 A063374 A063375 * A063377 A063378 A063379 KEYWORD easy,nonn AUTHOR Henry Bottomley, Jul 14 2001 EXTENSIONS Entry rewritten by N. J. A. Sloane Jan 23 2006 Definition corrected to a(-1) = 1 by Harry J. Smith, Aug 20 2009 STATUS approved

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Last modified October 13 18:57 EDT 2019. Contains 327981 sequences. (Running on oeis4.)