A291148 Expansion of 1 - x/(1 - x^2/(1 - x^3/(1 - x^4/(1 - x^5/(1 - x^6/(1 - ... - x^n/(1 - ...))))))), a continued fraction.

1, -1, 0, -1, 0, -1, -1, -1, -2, -2, -4, -4, -7, -9, -13, -19, -25, -38, -51, -75, -104, -149, -211, -298, -426, -600, -857, -1211, -1724, -2444, -3471, -4930, -6995, -9940, -14104, -20038, -28444, -40397, -57362, -81453, -115675, -164250, -233262, -331227

Offset: 0

Author

Seiichi Manyama, Aug 18 2017

Keywords

• sign

Comments

The sequence b(n>=1) = 1, 0, 1, 0, 1, 1, 1, 2, 2, 4, 4, 7, ... of absolute values counts fountains of n coins that cannot be separated into two or more fountains by cutting vertically through the fountain without splitting a coin. (This separation requires that the fountain is a left-right sequence of more elementary fountains counted by b(n).) A005169(n) = Sum_{compositions n=n1+n2+n3+...} Product b(n1)*b(n2)*.... - R. J. Mathar, Aug 22 2018

Examples

			G.f. = 1 - x - x^3 - x^5 - x^6 - x^7 - 2*x^8 - 2*x^9 - ...
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A005169, A290975, A291147.

Formula

Convolution inverse of A005169.

a(n) ~ c * d^n, where d = 1.42009048763893649946106129818306075366296460727614... and c = -0.093433697175825717154301151812109730023054876584907211486145769... - Vaclav Kotesovec, Oct 16 2017