A180967 Number of n-game win/loss series that contain at least one dead game.

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

Offset: 1

Author

Dmitry Kamenetsky, Jan 28 2011

Keywords

• nonn

Comments

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).

Examples

			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.
		

Crossrefs

See A181618 for win/loss/draw series.

Programs

Mathematica

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 *)

Formula

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