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, 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

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

Paolo Xausa, Table of n, a(n) for n = 1..2000

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

Crossrefs

A370826 are the corresponding denominators.

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

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

Sequence in context: A296544 A136635 A062743 * A009084 A096057 A179984

Adjacent sequences: A370822 A370823 A370824 * A370826 A370827 A370828

Keyword

nonn,frac,easy

Author

Hugo Pfoertner, Mar 08 2024

Status

approved