|
%I #79 Sep 08 2022 08:45:03
%S 1,2,6,20,72,272,1056,4160,16512,65792,262656,1049600,4196352,
%T 16781312,67117056,268451840,1073774592,4295032832,17180000256,
%U 68719738880,274878431232,1099512676352,4398048608256,17592190238720,70368752566272
%N a(-1) = 1; for n >= 0, a(n) = 2^n + 4^n = 2^n*(1 + 2^n).
%C Counts closed walks of length 2n+2 at a vertex of the cyclic graph on 8 nodes C_8.
%C 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
%C Number of monic irreducible polynomials of degree 1 in GF(2^n)[x,y]. - _Max Alekseyev_, Jan 23 2006
%C 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
%C With offset 0 = binomial transform of A122983. - _Gary W. Adamson_, Apr 18 2011
%D 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.
%H Harry J. Smith, <a href="/A063376/b063376.txt">Table of n, a(n) for n = -1..200</a>
%H M. Archibald, A. Blecher, A. Knopfmacher, M. E. Mays, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Archibald/arch3.html">Inversions and Parity in Compositions of Integers</a>, J. Int. Seq., Vol. 23 (2020), Article 20.4.1.
%H Georgia Benkart, Dongho Moon, <a href="http://arxiv.org/abs/1610.07837">Walks on Graphs and Their Connections with Tensor Invariants and Centralizer Algebras</a>, arXiv preprint arXiv:1610.07837 [math.RT], 2016-2017.
%H J. Brunvoll, S. J. Cyvin and B. N. Cyvin, <a href="http://dx.doi.org/10.1016/0022-2860(95)09039-8">Isomer enumeration of some polygonal systems representing polycyclic conjugated hydrocarbons</a>, J. Molec. Struct. (Theochem), 364 (1996), 1-13. (See Table 11.)
%H S. Capparelli, A. Del Fra, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Capparelli/cap3.html">Dyck Paths, Motzkin Paths, and the Binomial Transform</a>, Journal of Integer Sequences, 18 (2015), #15.8.5.
%H B. N. Cyvin et al., <a href="/A323850/a323850.png">Isomer enumeration of unbranched catacondensed polygonal systems with pentagons and heptagons</a>, 1996 [Annotated scanned copy of pages 118, 119 only].
%H T. A. Gulliver, <a href="http://www.m-hikari.com/ijcms/ijcms-2012/37-40-2012/gulliverIJCMS37-40-2012.pdf">Sums of Powers of Integers Divisible by Three</a>, Int. J. Contemp. Math. Sciences, Vol. 7, 2012, no. 38, 1895 - 1901.
%H D. Suprijanto and Rusliansyah, <a href="http://dx.doi.org/10.12988/ams.2014.4140">Observation on Sums of Powers of Integers Divisible by Four</a>, Applied Mathematical Sciences, Vol. 8, 2014, no. 45, 2219 - 2226.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-8).
%F a(n) = Sum_{k=0..n} if((n-k) mod 4 = 0, binomial(n, 2*k), 0)}. - _Paul Barry_, Sep 19 2005
%F a(n) = 4*a(n-1) - 2^n = 6*a(n-1) - 8*a(n-2) = A001576(n) - 1 = 2*A007582(n) = A005418(2*n+2) = A002378(A000079(n)).
%F G.f.: 1/x + (2-6*x)/((1-2*x)*(1-4*x)).
%F a(n) = ceiling(2^n*(2^n + 1)), n >= -1. - _Zerinvary Lajos_, Jan 07 2008
%F E.g.f.: exp(2*x)*cosh(x)^2. - _Paul D. Hanna_, Oct 25 2012
%F 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
%e a(1)=6 counts six round trips from, say, vertex no 1: 12121, 18181, 12181, 18121, 12321 and 18781. - _Wolfdieter Lang_, Nov 08 2011
%p seq(ceil(2^n*(2^n + 1)),n=-1..23); # _Zerinvary Lajos_, Jan 07 2008
%t Table[2^n + 4^n, {n, 0, 25}]
%o (PARI) a(n)={if(n>=0, 2^n*(1 + 2^n), 1)} \\ _Harry J. Smith_, Aug 20 2009
%o (PARI) {a(n)=n!*polcoeff((1 + exp(2*x +x*O(x^n)))^2/4,n)} \\ _Paul D. Hanna_, Oct 25 2012
%o (Magma) [1] cat [2^n + 4^n : n in [0..30]]; // _Wesley Ivan Hurt_, Jul 03 2020
%Y Cf. A000051, A006516, A007582, A034472, A034474, A034491, A052539, A062394, A062395, A062396, A007689, A063376, A063481, A074600 - A074624, A122983.
%Y A column of A323850.
%Y Essentially the same as A028402.
%K easy,nonn
%O -1,2
%A _Henry Bottomley_, Jul 14 2001
%E Entry rewritten by _N. J. A. Sloane_ Jan 23 2006
%E Definition corrected to a(-1) = 1 by _Harry J. Smith_, Aug 20 2009
|
