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.
Original entry on oeis.org
2, 16, 104, 128, 3872, 3328, 139904, 167936, 5038592, 2748416, 7886848, 2392064, 6530342912, 39182073856, 235092475904, 16594763776, 8463329656832, 381804347392, 304679869743104, 6647560798208, 10968475319140352, 2861341387784192, 8401385351544832, 5207012459675648
Offset: 1
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.
A370824 are the corresponding denominators.
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Array[Numerator[(3^#-1)/((3/2)^#-1)/2] &, 35] (* Paolo Xausa, Mar 13 2024 *)
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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))
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from math import gcd
def A370823(n): return (a:=3**n-1<Chai Wah Wu, Mar 07 2024
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.
Original entry on oeis.org
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
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.
A370826 are the corresponding denominators.
A052548(n+1)/3 is the ratio of winning probabilities when the odds are 2:1:1.
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Array[Numerator[3/4*(3^#-1)/(2^#-1)] &, 50] (* Paolo Xausa, Mar 11 2024 *)
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a370825(n) = numerator((3/4) * (3^n - 1) / (2^n - 1));
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from math import gcd
def A370825(n): return (a:=3**(n+1)-3>>1)//gcd(a,(1<Chai Wah Wu, Mar 10 2024
A370826
a(n) are the denominators corresponding to A370825(n).
Original entry on oeis.org
2, 1, 14, 1, 62, 3, 254, 17, 1022, 31, 178, 3, 16382, 5461, 65534, 257, 262142, 657, 1048574, 1271, 4194302, 60787, 356962, 12291, 67108862, 22369621, 268435454, 617093, 1073741822, 149943, 4294967294, 16843009, 746950834, 5726623061, 967879954, 981, 274877906942
Offset: 1
A370825(n)/a(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.
A370825 are the corresponding numerators.
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Array[Denominator[3/4*(3^#-1)/(2^#-1)] &, 50] (* Paolo Xausa, Mar 11 2024 *)
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a370826(n) = denominator((3/4) * (3^n - 1) / (2^n - 1));
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from math import gcd
def A370826(n): return (a:=(1<>1) # Chai Wah Wu, Mar 10 2024
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.
Original entry on oeis.org
3, 63, 1053, 16443, 250533, 28431, 56859813, 853737003, 12811093893, 17472421953, 2883131020773, 25013333547, 648727335888453, 9730949220408843, 145964473398624933, 128792265384372219, 32842036136344638213, 3703989419737954191, 7389459197057088616293, 10076535434903231752353
Offset: 1
a(n)/A370828(n) for n = 1..8: 3/2, 63/32, 1053/392, 16443/4352, 250533/46112, 28431/3584, 56859813/4860032, 853737003/49160192.
A370828 are the corresponding denominators.
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a370827(n) = numerator((2/3) * (3/5)^n * ((5/2)^n - 1) / (1 - (3/5)^n))
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from math import gcd
def A370827(n): return (a:=3**(n-1)*(5**n-(1<Chai Wah Wu, Mar 12 2024
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