A366567 a(n) is the mode of the probability distributions from which the expected game lengths in A366566 were determined.
2, 3, 4, 7, 10, 11, 16, 19, 22, 27, 32, 37, 42, 47, 54, 59
Offset: 1
This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
4.5597944999598458154817364845572481176367423801661405635...
First[RealDigits[Gamma[1/3]^9/(2Pi)^4,10,100]] (* Paolo Xausa, Oct 14 2023 *)
sqrt(Pi)/(3*sqrt(3))*(gamma(1/3)/gamma(5/6))^3
from itertools import permutations
def T(n):
nodes = [(i,j) for i in range(n+2) for j in range((n+2-i)//2+1)]
m = len(nodes)
Q0 = {x:{y:0 for y in nodes} for x in nodes}
for x in nodes:
c1 = x+(n+2-sum(x),)
for i,j in permutations(range(3),int(2)):
if c1[i] and c1[j]:
c2 = list(c1)
c2[i] -= 1
c2[j] += 1
y = (c2[0],min(c2[1:]))
if c2[0] != n+2:
Q0[x][y] += n+2-c2[0]
Q0 = matrix(QQ,[list(R.values()) for R in Q0.values()])
s = sum(Q0.columns())
Q = identity_matrix(QQ,m-1)
for i in range(1,m):
for j in range(1,m):
if s[i] != 0: Q[i-1,j-1] -= Q0[i,j]/s[i]
return (Q**(-1)*ones_matrix(QQ,m-1))[-2,0]
def A366995(n):
return T(n).numerator()
def A366996(n):
return T(n).denominator()
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