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%I #14 Jun 13 2015 00:50:30
%S 1,1,4,7,20,39,96,191,432,863,1856,3711,7744,15487,31744,63487,128768,
%T 257535,519168,1038335,2085888,4171775,8364032,16728063,33501184,
%U 67002367,134103040,268206079,536625152,1073250303,2146959360
%N Winning binary "same game" templates of length n as defined below.
%C 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.
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (2,5,-10,-8,16,4,-8).
%F a(2*n-1)= 2^(2*n-1) -n * 2^(n-1), a(2*n)= 2*a(2*n-1) -1.
%F 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
%e 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.
%Y a(2*n-1)= A008353(n-1), cf. A035615, A007931, A066067.
%K nonn,easy
%O 1,3
%A _Frank Ellermann_, Dec 23 2001
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