|
|
A063376
|
|
a(-1) = 1; for n >= 0, a(n) = 2^n + 4^n = 2^n*(1 + 2^n).
|
|
60
|
|
|
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
|
|
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
|
|
|
FORMULA
|
a(n) = Sum_{k=0..n} if((n-k) mod 4 = 0, binomial(n, 2*k), 0)}. - Paul Barry, Sep 19 2005
G.f.: 1/x + (2-6*x)/((1-2*x)*(1-4*x)).
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
|
|
|
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.
|
|
KEYWORD
|
easy,nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|