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A063376 a(-1) = 1; for n >= 0, a(n) = 2^n + 4^n = 2^n*(1+2^n). 56
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. [From Jaroslav Krizek, Aug 14 2009]

With offset 0 = binomial transform of A122983. [Gary W. Adamson, Apr 18 2011]

REFERENCES

S. Capparelli, A. Del Fra, Dyck Paths, Motzkin Paths, and the Binomial Transform, Journal of Integer Sequences, 18 (2015), #15.8.5.

D. Suprijanto and Rusliansyah, Observation on Sums of Powers of Integers Divisible by Four, Applied Mathematical Sciences, Vol. 8, 2014, no. 45, 2219 - 2226; http://dx.doi.org/10.12988/ams.2014.4140.

LINKS

Harry J. Smith, Table of n, a(n) for n=-1,200

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.)

T. A. Gulliver, Sums of Powers of Integers Divisible by Three, Int. J. Contemp. Math. Sciences, Vol. 7, 2012, no. 38, 1895 - 1901.

Index entries for linear recurrences with constant coefficients, signature (6,-8).

FORMULA

a(n) = sum{k=0..n, if(mod(n-k, 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) = ceil(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) { for (n = -1, 200, if (n>=0, p=2^n; a=p*(1 + p), a=1); write("b063376.txt", n, " ", a) ) } [From 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.

Sequence in context: A049141 A049129 * A161168 A049139 A071356 A141200

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 May 22 19:31 EDT 2017. Contains 286885 sequences.