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A180967
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Number of n-game win/loss series that contain at least one dead game.
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1
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0, 0, 4, 4, 20, 24, 88, 116, 372, 520, 1544, 2248, 6344, 9520, 25904, 39796, 105332, 164904, 427048, 679064, 1727640, 2783440, 6977744, 11368904, 28146120, 46307664, 113416528, 188202256, 456637712, 763506784
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OFFSET
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1,3
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COMMENTS
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A series of n games are played between two teams. The outcome of each game is either a win or a loss (there are no draws). A team wins the whole series if it wins k=floor(n/2)+1 games or more. If a team reaches k wins then the games that follow (if there are any) are dead games, because their outcome cannot affect the outcome of the series.
Number of n-game series whose outcome is decided in the last game is A063886(n).
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LINKS
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FORMULA
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The last game is "alive" if and only if the result of the first n-1 games
is either (if n is odd) (n-1)/2 wins for both teams, or (if n is even) n/2 wins for one and n/2-1 for the other. Hence a(n)=2^n - 2C(n-1,(n-1)/2) for odd n and a(n)=2^n - 4C(n-1,n/2) for even n. - Robert Israel, Jan 28 2011
-n*a(n) +n*a(n-1) +2*(3*n-5)*a(n-2) +4*(-n+1)*a(n-3) +8*(-n+4)*a(n-4)=0. - R. J. Mathar, May 19 2014
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EXAMPLE
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We can represent an n-game series as a binary string of length n, where '0' means a loss for the first team and '1' means a win for the first team. For n=3 there are 2^3=8 possible game series. Out of these there are 4 that contain at least one dead game (the last one): 000, 001, 110, 111. Hence a(3)=4.
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MATHEMATICA
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f[n_] := 2^n - 2*If[ OddQ@ n, Binomial[n - 1, (n - 1)/2], 2 Binomial[n - 1, n/2]]; Array[f, 30] (* Robert G. Wilson v *)
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CROSSREFS
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See A181618 for win/loss/draw series.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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