A285903 G.f.: 1 + Sum_{n>=1} a(n)*x^n/(1 - x^n) = 1/(1 - x/(1 - x^2/(1 - x^3/(1 - x^4/(1 - ...))))).

1, 0, 1, 2, 4, 7, 14, 23, 43, 73, 134, 223, 405, 689, 1216, 2094, 3678, 6333, 11080, 19152, 33363, 57798, 100549, 174262, 302898, 525328, 912448, 1583069, 2748892, 4769842, 8281087, 14371045, 24946819, 43295806, 75153267, 130434130, 226401111, 392944875, 682038592, 1183770424, 2054659668, 3566162246

Offset: 1

Author

Ilya Gutkovskiy, Apr 28 2017

Keywords

• nonn

Examples

			G.f.: 1 + x/(1 - x) + x^3/(1 - x^3) + 2*x^4/(1 - x^4) + 4*x^5/(1 - x^5) + ... = 1/(1 - x/(1 - x^2/(1 - x^3/(1 - x^4/(1 - ...))))).
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..1000
• Eric Weisstein's World of Mathematics, Rogers-Ramanujan Continued Fraction
• N. J. A. Sloane, Transforms

Crossrefs

Cf. A005169, A192728.

Programs

Mathematica

nn = 42; f[x_] := 1 + Sum[a[n] x^n/(1 - x^n), {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - 1/(1 + ContinuedFractionK[-x^n, 1, {n, 1, nn}]), {x, 0, nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten

Formula

G.f.: 1 + Sum_{n>=1} a(n)*x^n/(1 - x^n) = (Sum_{n>=0} (-1)^n*x^(n*(n+1)) /Product_{k=1..n} (1 - x^k)) / (Sum_{n>=0} (-1)^n*x^(n^2)/Product_{k=1..n} (1 - x^k)).

Sum_{d|n} a(d) = A005169(n) for n > 0.

From Vaclav Kotesovec, Apr 30 2017: (Start)

a(n) ~ c * d^n, where

d = 1/A347901 = 1.735662824530347425658260749719668530254652847290392754609934...

c = 0.31236332459674145306627970724066492149823012868471473538681348971946...

(End)