login
A333155
Decimal expansion of a constant related to the asymptotics of A268188 and A333153.
7
5, 9, 3, 2, 4, 2, 2, 1, 5, 0, 0, 3, 3, 6, 9, 1, 2, 7, 1, 8, 4, 1, 3, 7, 6, 1, 7, 3, 3, 0, 2, 5, 5, 9, 5, 4, 1, 1, 0, 9, 9, 5, 9, 5, 4, 9, 6, 2, 7, 9, 5, 7, 4, 2, 9, 0, 6, 0, 2, 4, 5, 7, 8, 6, 0, 4, 5, 3, 5, 9, 2, 2, 3, 8, 5, 4, 6, 8, 1, 3, 3, 3, 3, 2, 5, 5, 0, 4, 8, 0, 7, 2, 0, 2, 8, 1, 9, 6, 6, 3, 9, 7, 1, 0, 7, 1
OFFSET
0,1
FORMULA
Equals sqrt(15) * log(phi) / Pi, where phi = A001622 = (1+sqrt(5))/2 is the golden ratio.
If m >= 0 and g.f. is Sum_{k>=1} (k^m * x^(k^2) / Product_{j=1..k} (1 - x^j)), then a(n) ~ A333155^m * phi^(1/2) * exp(2*Pi*sqrt(n/15)) * n^((2*m-3)/4) / (2 * 3^(1/4) * 5^(1/2)).
If m >= 0 and g.f. is Sum_{k>=1} (k^m * x^(k*(k+1)) / Product_{j=1..k} (1 - x^j)), then a(n) ~ A333155^m * exp(2*Pi*sqrt(n/15)) * n^((2*m-3)/4) / (2 * 3^(1/4) * 5^(1/2) * phi^(1/2)).
EXAMPLE
0.5932422150033691271841376173302559541109959549627957429060245786...
MAPLE
evalf(sqrt(15) * log((sqrt(5) + 1)/2) / Pi, 120);
MATHEMATICA
RealDigits[Sqrt[15]*Log[GoldenRatio]/Pi, 10, 105][[1]]
KEYWORD
nonn,cons
AUTHOR
Vaclav Kotesovec, Mar 09 2020
STATUS
approved