OFFSET
1,3
COMMENTS
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.
The number of drawn n-game series is A002426(n).
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..200
FORMULA
From Andrew Howroyd, Jan 04 2020: (Start)
a(n) = 3^n - 3*A002426(n).
a(n) = 6*A055218(n-3) for n >= 3.
G.f.: 2 + 1/(1 - 3*x) - 3/sqrt(1 - 2*x - 3*x^2). (End)
EXAMPLE
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.
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Dmitry Kamenetsky, Jan 30 2011
EXTENSIONS
Terms a(16) and beyond from Andrew Howroyd, Jan 04 2020
STATUS
approved