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A063376 a(-1) = 1; for n >= 0, a(n) = 2^n + 4^n = 2^n*(1 + 2^n). 61
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

M. Archibald, A. Blecher, A. Knopfmacher, M. E. Mays, Inversions and Parity in Compositions of Integers, J. Int. Seq., Vol. 23 (2020), Article 20.4.1.

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, 2*k), 0)}. - Paul Barry, Sep 19 2005

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)).

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

(MAGMA) [1] cat [2^n + 4^n : n in [0..30]]; // Wesley Ivan Hurt, Jul 03 2020

CROSSREFS

Cf. A000051, A006516, A007582, A034472, A034474, A034491, A052539, A062394, A062395, A062396, A007689, A063376, A063481, A074600 - A074624, A122983.

A column of A323850.

Essentially the same as A028402.

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 23 20:03 EDT 2020. Contains 337975 sequences. (Running on oeis4.)