A188697 Expansion of (1+2*x^2)/(1-26*x+2*x^2-52*x^3+4*x^4).

1, 26, 676, 17576, 456972, 11881168, 308907672, 8031529376, 208817941280, 5429219088800, 141158464323104, 3670088041052160, 95421456259562432, 2480936209934965120, 64503778490067388160, 1677083603695215199744, 43603793136187040353536, 1133688727070116383116288, 29475649649828842801150464, 766360202350076625301264384

Offset

0,2

Comments

From the Noonan and Zeilberger link below, a(n) is the number of words in the 26-letter English alphabet {A,B,C,...,X,Y,Z} that do not contain any of the "bad" words: PIPI or CACA or PICA or CAPI. The expected wait time to see an occurrence of one of any of these four words is G(1/26) = 114582. The expected wait time to see all four of these words is 979223595402/1028195 (approximately 952371). - Geoffrey Critzer, May 17 2014

Links

Table of n, a(n) for n=0..19.

J. Noonan and D. Zeilberger, The Goulden-Jackson cluster method: extensions, applications and implementations

Index entries for linear recurrences with constant coefficients, signature (26,-2,52,-4).

Formula

G.f.: (1+2*x^2)/(1-26*x+2*x^2-52*x^3+4*x^4).

a(0)=1, a(1)=26, a(2)=676, a(3)=17576, a(n)=26*a(n-1)-2*a(n-2)+ 52*a(n-3)- 4*a(n-4). - Harvey P. Dale, Oct 04 2014

Mathematica

sol=Solve[{A==-z^4-z^2A-z^2D, B==-z^4-z^2B-z^2C, C==-z^4-z^2A-z^2D, D==-z^4-z^2B-z^2C}, {A, B, C, D}]; nn=20; CoefficientList[Series[1/(1-26z-A-B-C-D)/.sol, {z, 0, nn}], z] (* Geoffrey Critzer, May 17 2014 *)
LinearRecurrence[{26, -2, 52, -4}, {1, 26, 676, 17576}, 30] (* Harvey P. Dale, Oct 04 2014 *)

Crossrefs

Cf. A188696.

Sequence in context: A206695 A171300 A360509 * A188696 A009970 A041313

Adjacent sequences: A188694 A188695 A188696 * A188698 A188699 A188700

Keyword

nonn,easy

Author

N. J. A. Sloane, Apr 08 2011

Status

approved