A066345 Winning binary "same game" templates of length n as defined below.

1, 1, 4, 7, 20, 39, 96, 191, 432, 863, 1856, 3711, 7744, 15487, 31744, 63487, 128768, 257535, 519168, 1038335, 2085888, 4171775, 8364032, 16728063, 33501184, 67002367, 134103040, 268206079, 536625152, 1073250303, 2146959360

Offset

1,3

Comments

A "same game template" is a pattern representing the run pattern of a string in a 2 symbol alphabet. Each position in the template represents either an isolated symbol, or a run of two or more identical symbols. Such a template can be represented as a ternary number without digit 0 (A007931), where 2 represents any run of 2 or more identical symbols and ternary 1 represents remaining single bitsymbols, e.g. 211 for 0010, 1101, 00010, etc. A winning template represents an infinite subset of winning binary "same games", e.g. 121 for 0110, 1001, 01110, etc.

Links

Table of n, a(n) for n=1..31.

Index entries for linear recurrences with constant coefficients, signature (2,5,-10,-8,16,4,-8).

Formula

a(2*n-1)= 2^(2*n-1) -n * 2^(n-1), a(2*n)= 2*a(2*n-1) -1.

G.f. x*( 1-x-3*x^2+4*x^3+4*x^4-4*x^5 ) / ( (x-1)*(2*x-1)*(1+x)*(-1+2*x^2)^2 ). - R. J. Mathar, May 07 2013

Example

There are a(3)= 4 winning templates 121, 122, 221, 222 with 3 ternary digits and a(4)= 7 winning templates 1212, 2121, 1222, 2221, 2122, 2212, 2222.

Crossrefs

a(2*n-1)= A008353(n-1), cf. A035615, A007931, A066067.

Sequence in context: A266822 A296624 A296663 * A355357 A026570 A111955

Adjacent sequences: A066342 A066343 A066344 * A066346 A066347 A066348

Keyword

nonn,easy

Author

Frank Ellermann, Dec 23 2001

Status

approved