A370825 - OEIS

%I #13 Mar 11 2024 10:15:05

%S 3,2,39,4,363,26,3279,328,29523,1342,11553,292,2391483,1195742,

%T 21523359,126608,193710243,728234,1743392199,3169804,15690529803,

%U 341098474,3004569537,155181064,1270932914163,635466457082,11438396227479,39442745612,102945566047323,21563796826

%N 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.

%C 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.

%H Paolo Xausa, <a href="/A370825/b370825.txt">Table of n, a(n) for n = 1..2000</a>

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

%e 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.

%t Array[Numerator[3/4*(3^#-1)/(2^#-1)] &, 50] (* _Paolo Xausa_, Mar 11 2024 *)

%o (PARI) a370825(n) = numerator((3/4) * (3^n - 1) / (2^n - 1));

%o (Python)

%o from math import gcd

%o def A370825(n): return (a:=3**(n+1)-3>>1)//gcd(a,(1<<n+1)-2) # _Chai Wah Wu_, Mar 10 2024

%Y A370826 are the corresponding denominators.

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

%Y Cf. A370823, A370824 for odds 2:1:0.

%K nonn,frac,easy

%O 1,1

%A _Hugo Pfoertner_, Mar 08 2024