A181618 Number of n-game win/loss/draw series that contain at least one dead game.
0, 0, 6, 24, 90, 306, 1008, 3240, 10266, 32190, 100188, 310074, 955500, 2934288, 8986086, 27456840, 83735370, 254962062, 775270908, 2354646294, 7144301016, 21657653028, 65603494458, 198584527338, 600758100540, 1816426149876, 5489387016378, 16582071393240
Offset: 1
Keywords
Examples
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.
Links
- Andrew Howroyd, Table of n, a(n) for n = 1..200
Programs
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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
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)
Extensions
Terms a(16) and beyond from Andrew Howroyd, Jan 04 2020
Comments