

A370825


a(n) is the numerator of the ratio of winning probabilities in a game similar to A370823, but with a draw and single round odds A:B:draw of 3:2:1.


3



3, 2, 39, 4, 363, 26, 3279, 328, 29523, 1342, 11553, 292, 2391483, 1195742, 21523359, 126608, 193710243, 728234, 1743392199, 3169804, 15690529803, 341098474, 3004569537, 155181064, 1270932914163, 635466457082, 11438396227479, 39442745612, 102945566047323, 21563796826
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,1


COMMENTS

Such a game can be implemented, for instance, by rolling a single die per round, with A winning the round on numbers 1, 2, 3, B winning on 4, 5 and a draw on 6. To win the game it is necessary to win n rounds in a row. The draw also terminates winning streaks of A or B.


LINKS



FORMULA

a(n)/A370826(n) = (3/4) * (3^n  1) / (2^n  1).


EXAMPLE

a(n)/A370826(n) for n=1..14: 3/2, 2, 39/14, 4, 363/62, 26/3, 3279/254, 328/17, 29523/1022, 1342/31, 11553/178, 292/3, 2391483/16382, 1195742/5461.


MATHEMATICA

Array[Numerator[3/4*(3^#1)/(2^#1)] &, 50] (* Paolo Xausa, Mar 11 2024 *)


PROG

(PARI) a370825(n) = numerator((3/4) * (3^n  1) / (2^n  1));
(Python)
from math import gcd


CROSSREFS

A370826 are the corresponding denominators.
A052548(n+1)/3 is the ratio of winning probabilities when the odds are 2:1:1.


KEYWORD

nonn,frac,easy


AUTHOR



STATUS

approved



