A225435 Numerators of convergents to the general continued fraction 1/(1 + 2/(1 + 3/(1 + 4/(1+ ...)))).

1, 1, 2, 4, 7, 19, 68, 44, 416, 758, 6092, 24284, 10348, 110864, 997828, 4545476, 827252, 5166356, 255994804, 1289266004, 3332578444, 8757252244, 3766552348, 27079574548, 1434303566956, 4061479240156, 46849154788124, 54858398447372, 816458740546228, 189647639388428

Offset: 1

Author

Eric W. Weisstein, May 07 2013

Keywords

• nonn
• frac

Examples

			1, 1/3, 2/3, 4/9, 7/12, 19/39, ... = A225435(n)/A225436(n).
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..843
• Eric Weisstein's World of Mathematics, Continued Fraction Constants
• Eric Weisstein's World of Mathematics, Generalized Continued Fraction

Crossrefs

Cf. A225436 (denominators).

Cf. A111129 (decimal digits of infinite c.f.).

Related to A000932.

Programs

Mathematica

Numerator[Table[ContinuedFractionK[k, 1, {k, 1, n}], {n, 30}]]

Formula

E.g.f.: (1/2)*(-2+e^((1/2)*z*(2+z))*(1+z)(2+sqrt(2*e*Pi)*erf(1/sqrt(2)))-e^((1/2)*(1+z)^2)*sqrt(2*Pi)*(1+z)*erf((1+z)/sqrt(2))).

Lim_{n->infinity} A225435(n)/A225436(n) = sqrt(2/(e*Pi))/erfc(1/sqrt(2))-1 = A111129.