A333155 Decimal expansion of a constant related to the asymptotics of A268188 and A333153.

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

Links

Table of n, a(n) for n=0..105.

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]]

Crossrefs

Cf. A003114, A268188, A333141, A333151, A333152.

Cf. A003106, A333153, A333154.

Sequence in context: A255681 A021948 A306883 * A256500 A239545 A228402

Adjacent sequences: A333152 A333153 A333154 * A333156 A333157 A333158

Keyword

nonn,cons

Author

Vaclav Kotesovec, Mar 09 2020

Status

approved