A370827 a(n) is the numerator of the ratio of winning probabilities P_A/P_B of winning in a 2-player game with a ratio of odds for A and B in a single round of 3:2. To win the game it is necessary to win n rounds in a row.

3, 63, 1053, 16443, 250533, 28431, 56859813, 853737003, 12811093893, 17472421953, 2883131020773, 25013333547, 648727335888453, 9730949220408843, 145964473398624933, 128792265384372219, 32842036136344638213, 3703989419737954191, 7389459197057088616293, 10076535434903231752353

Offset

1,1

Comments

If you play the game with odds of 3:2 with one dice, you can decide what happens to the unused number. In the present sequence, this number is ignored without effect. However, if one defines the occurrence of the 6th number as a draw, which interrupts consecutive wins by both players, then the sequence pair A370825/A370826 results. The addition of a draw increases the odds ratio in favor of the player who has the higher chance of winning a single round. The relative advantage is small with 2 games to be won, i.e., 2/(63/32)-1 = 1/63, with 3 rounds 1/27, with 4 rounds 965/16443, but increases with a higher number of consecutive rounds to be won, and reaches asymptotically 1/8.

Links

Table of n, a(n) for n=1..20.

Formula

a(n)/A370828(n) = (2/3) * (3/5)^n * ((5/2)^n - 1) / (1 - (3/5)^n).

Example

a(n)/A370828(n) for n = 1..8: 3/2, 63/32, 1053/392, 16443/4352, 250533/46112, 28431/3584, 56859813/4860032, 853737003/49160192.

Prog

(PARI) a370827(n) = numerator((2/3) * (3/5)^n * ((5/2)^n - 1) / (1 - (3/5)^n))
(Python)
from math import gcd
def A370827(n): return (a:=3**(n-1)*(5**n-(1<<n)))//gcd(a, 5**n-3**n<<n-1) # Chai Wah Wu, Mar 12 2024

Crossrefs

A370828 are the corresponding denominators.

Cf. A370823, A370824, A370825, A370826.

Sequence in context: A151993 A120053 A139293 * A133275 A258657 A331012

Adjacent sequences: A370824 A370825 A370826 * A370828 A370829 A370830

Keyword

nonn,frac,easy

Author

Hugo Pfoertner, Mar 09 2024

Status

approved