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A066345 Winning binary "same game" templates of length n as defined below. 3
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 (list; graph; refs; listen; history; text; internal format)
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 * A026570 A111955 A244791

Adjacent sequences:  A066342 A066343 A066344 * A066346 A066347 A066348

KEYWORD

nonn,easy

AUTHOR

Frank Ellermann, Dec 23 2001

STATUS

approved

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Last modified July 24 06:56 EDT 2021. Contains 346273 sequences. (Running on oeis4.)