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A181618 Number of n-game win/loss/draw series that contain at least one dead game.

%I A181618 #16 Jan 05 2020 05:39:25
%S A181618 0,0,6,24,90,306,1008,3240,10266,32190,100188,310074,955500,2934288,
%T A181618 8986086,27456840,83735370,254962062,775270908,2354646294,7144301016,
%U A181618 21657653028,65603494458,198584527338,600758100540,1816426149876,5489387016378,16582071393240
%N A181618 Number of n-game win/loss/draw series that contain at least one dead game.
%C A181618 A series of n games are played between two teams. The outcome of each game is either a win, a loss or a draw. A team wins the whole series if it has more wins than its opponent. If the difference between the number of wins of each team is k and there are less than k games remaining in the series, then the remaining games are dead. The outcome of the dead games cannot affect the outcome of the series.
%C A181618 The number of drawn n-game series is A002426(n).
%H A181618 Andrew Howroyd, <a href="/A181618/b181618.txt">Table of n, a(n) for n = 1..200</a>
%F A181618 From _Andrew Howroyd_, Jan 04 2020: (Start)
%F A181618 a(n) = 3^n - 3*A002426(n).
%F A181618 a(n) = 6*A055218(n-3) for n >= 3.
%F A181618 G.f.: 2 + 1/(1 - 3*x) - 3/sqrt(1 - 2*x - 3*x^2). (End)
%e A181618 We can represent an n-game series as a ternary string of length n, where '0' means a draw, '1' means loss for the first team and '2' means a win for the first team. For n=3 there are 3^3=27 possible game series. Out of these there are 6 that contain at least one dead game (the last one): 110, 111, 112, 220, 221, 222. Hence a(3)=6.
%o A181618 (PARI) concat([0,0], Vec(2 + 1/(1 - 3*x) - 3/sqrt(1 - 2*x - 3*x^2 + O(x^30)))) \\ _Andrew Howroyd_, Jan 04 2020
%Y A181618 See A180967 for win/loss series.
%Y A181618 Cf. A002426, A055218.
%K A181618 nonn
%O A181618 1,3
%A A181618 _Dmitry Kamenetsky_, Jan 30 2011
%E A181618 Terms a(16) and beyond from _Andrew Howroyd_, Jan 04 2020