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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.

%I A370825 #13 Mar 11 2024 10:15:05
%S A370825 3,2,39,4,363,26,3279,328,29523,1342,11553,292,2391483,1195742,
%T A370825 21523359,126608,193710243,728234,1743392199,3169804,15690529803,
%U A370825 341098474,3004569537,155181064,1270932914163,635466457082,11438396227479,39442745612,102945566047323,21563796826
%N 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.
%C A370825 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 A370825 Paolo Xausa, <a href="/A370825/b370825.txt">Table of n, a(n) for n = 1..2000</a>
%F A370825 a(n)/A370826(n) = (3/4) * (3^n - 1) / (2^n - 1).
%e A370825 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 A370825 Array[Numerator[3/4*(3^#-1)/(2^#-1)] &, 50] (* _Paolo Xausa_, Mar 11 2024 *)
%o A370825 (PARI) a370825(n) = numerator((3/4) * (3^n - 1) / (2^n - 1));
%o A370825 (Python)
%o A370825 from math import gcd
%o A370825 def A370825(n): return (a:=3**(n+1)-3>>1)//gcd(a,(1<<n+1)-2) # _Chai Wah Wu_, Mar 10 2024
%Y A370825 A370826 are the corresponding denominators.
%Y A370825 A052548(n+1)/3 is the ratio of winning probabilities when the odds are 2:1:1.
%Y A370825 Cf. A370823, A370824 for odds 2:1:0.
%K A370825 nonn,frac,easy
%O A370825 1,1
%A A370825 _Hugo Pfoertner_, Mar 08 2024