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

Author

Vaclav Kotesovec, Mar 09 2020

Keywords

• nonn
• cons

Examples

			0.5932422150033691271841376173302559541109959549627957429060245786...
		

Crossrefs

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

Cf. A003106, A333153, A333154.

Programs

Maple

evalf(sqrt(15) * log((sqrt(5) + 1)/2) / Pi, 120);

Mathematica

RealDigits[Sqrt[15]*Log[GoldenRatio]/Pi, 10, 105][[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)).