login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A370823 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 2:1. To win the game it is necessary to win n rounds in a row. 6
2, 16, 104, 128, 3872, 3328, 139904, 167936, 5038592, 2748416, 7886848, 2392064, 6530342912, 39182073856, 235092475904, 16594763776, 8463329656832, 381804347392, 304679869743104, 6647560798208, 10968475319140352, 2861341387784192, 8401385351544832, 5207012459675648 (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 if the numbers are 1 to 4 and B winning if the numbers are 5 and 6.
LINKS
IBM Research, A Dice Game, Ponder This Challenge, February 2024.
FORMULA
See the solution page of the "Ponder This" challenge for the formula derived from the Markov matrix representing the rules of the game.
Numerator of 2^(n-1)*(3^n-1)/(3^n-2^n). - Chai Wah Wu, Mar 07 2024
EXAMPLE
a(n)/A370824(n) for n = 1..11: 2/1, 16/5, 104/19, 128/13, 3872/211, 3328/95, 139904/2059, 167936/1261, 5038592/19171, 2748416/5275, 7886848/7613.
MATHEMATICA
Array[Numerator[(3^#-1)/((3/2)^#-1)/2] &, 35] (* Paolo Xausa, Mar 13 2024 *)
PROG
(PARI) a370823_4(n, A=2/3, B=1/3) = my (an=A^n, bn=B^n); (1-A) * an * (1-bn) / ((1-B) * bn * (1-an));
\\ or by determination of the eigenvalues of the Markov matrix
a370823_4(n, na=2, nb=1) = { if (n==1, na/nb, my (ntot=na+nb, A=na/ntot, B=nb/ntot, M=matrix(2*n+1)); M[1, 2]=A; M[1, 3]=B; for (rp=1, n-1, my (rb=2*rp+1, ra=rb-1); M[ra, 3]=B; M[rb, 2]=A; M[ra, ra+2]=A; M[rb, rb+2]=B); M[2*n, 2*n]=M[2*n+1, 2*n+1]=1; my (ME=mateigen(M)); ME[1, 2]/ME[1, 3])};
numerator(a370823_4(n))
(Python)
from math import gcd
def A370823(n): return (a:=3**n-1<<n-1)//gcd(a, 3**n-(1<<n)) # Chai Wah Wu, Mar 07 2024
CROSSREFS
A370824 are the corresponding denominators.
Sequence in context: A207853 A208022 A207803 * A059204 A187248 A236958
KEYWORD
nonn,frac
AUTHOR
Hugo Pfoertner, Mar 02 2024
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified July 15 11:27 EDT 2024. Contains 374332 sequences. (Running on oeis4.)