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Numerators of convergents to the general continued fraction 1/(1 + 2/(1 + 3/(1 + 4/(1+ ...)))).
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%I #36 Oct 19 2021 21:58:35

%S 1,1,2,4,7,19,68,44,416,758,6092,24284,10348,110864,997828,4545476,

%T 827252,5166356,255994804,1289266004,3332578444,8757252244,3766552348,

%U 27079574548,1434303566956,4061479240156,46849154788124,54858398447372,816458740546228,189647639388428

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

%H Seiichi Manyama, <a href="/A225435/b225435.txt">Table of n, a(n) for n = 1..843</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ContinuedFractionConstants.html">Continued Fraction Constants</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GeneralizedContinuedFraction.html">Generalized Continued Fraction</a>

%F 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))).

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

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

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

%Y Cf. A225436 (denominators).

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

%Y Related to A000932.

%K nonn,frac

%O 1,3

%A _Eric W. Weisstein_, May 07 2013