A225436 -id:A225436 - cp's OEIS frontend

A111129 Decimal expansion of the continued fraction 1+1/(1+2/(1+3/(1+4/(1+5/(1+...))))).

1, 5, 2, 5, 1, 3, 5, 2, 7, 6, 1, 6, 0, 9, 8, 1, 2, 0, 9, 0, 8, 9, 0, 9, 0, 5, 3, 6, 3, 9, 0, 5, 7, 8, 7, 1, 3, 3, 0, 7, 1, 1, 6, 3, 6, 4, 9, 2, 0, 6, 0, 3, 3, 3, 5, 5, 4, 6, 3, 1, 3, 9, 4, 2, 4, 2, 7, 2, 2, 6, 9, 2, 5, 5, 0, 7, 9, 5, 0, 3, 1, 6, 8, 7, 0, 2, 2, 8, 0, 1, 1, 8, 2, 6, 7, 2, 1, 1, 6, 5, 5, 2, 1, 4, 0

Offset: 1

N. J. A. Sloane, based on correspondence from Tom Raes (tommy1729(AT)hotmail.com) and Steven Finch, Sep 22 2005

			1.52513527616098120908909053639057871330711636492060333554631394242...

B. C. Berndt, Y.-S. Choi and S.-Y. Kang, The problems submitted by Ramanujan to the Journal of the Indian Mathematical Society, Continued Fractions: From Analytic Number Theory to Constructive Approximation, ed. B. C. Berndt and F. Gesztesy, Amer. Math. Soc., 1999, pp. 15-56.
S. R. Finch, "Mathematical Constants", Cambridge, pp. 423-428.
H. S. Wall, Analytic Theory of Continued Fractions, Van Nostrand, 1948, pp. 356-358, 367

G. C. Greubel, Table of n, a(n) for n = 1..5000
Eric Weisstein's World of Mathematics, Continued Fraction Constants
Eric Weisstein's World of Mathematics, Generalized Continued Fraction

Cf. A099287, A108088, A111188.

Cf. A225435, A225436 (numerators and denominators of convergents to c.f.).

RealDigits[1/(Sqrt[Pi*E/2]*Erfc[1/Sqrt[2]]), 10, 111][[1]]

1/(sqrt(Pi*exp(1)/2)*erfc(1/sqrt(2))) \\ G. C. Greubel, Jan 24 2017

Equals the reciprocal of sqrt(pi*e/2)*erfc(1/sqrt(2)), where erfc is the complementary error function.

More terms from Robert G. Wilson v and Hans Havermann, Oct 17 2005

Definition corrected by Steven Finch, Feb 05 2009

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

Original entry on oeis.org

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

Eric W. Weisstein, May 07 2013

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

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

Cf. A225436 (denominators).
Cf. A111129 (decimal digits of infinite c.f.).
Related to A000932.

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

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.

A289491 a(n) = denominator of 1/(1 + 1/(1 + 2/(1 + ... (1 + n)))).

Original entry on oeis.org

2, 4, 5, 13, 19, 58, 191, 131, 1187, 2231, 17519, 71063, 29881, 323423, 2887921, 13237457, 2397389, 15030317, 742458253, 3748521653, 9670072483, 25451905333, 10932619111, 78684575461, 4163946939067, 11799518538967, 136025604432743, 159359728522979

Offset: 1

Seiichi Manyama, Sep 02 2017

			1/2, 3/4, 3/5, 9/13, 12/19, 39/58, 123/191, 87/131, 771/1187, 1473/2231, 11427/17519, 46779/71063, 19533/29881, ... = A225436/A289491 -> A108088.
A225436(1)/a(1) = 1/2  = 1/(1 + 1)                         =  1/2,
A225436(2)/a(2) = 3/4  = 1/(1 + 1/(1 + 2))                 =  3/4,
A225436(3)/a(3) = 3/5  = 1/(1 + 1/(1 + 2/(1 + 3)))         =  6/10,
A225436(4)/a(4) = 9/13 = 1/(1 + 1/(1 + 2/(1 + 3/(1 + 4)))) = 18/26.

Seiichi Manyama, Table of n, a(n) for n = 1..843
OEIS Wiki, A remarkable formula of Ramanujan

Cf. A000085, A000932, A108088, A225435, A225436 (numerators).

p:= (i, n)-> `if`(i=n, (1+n), 1+i/p(i+1,n)):
a:= n-> denom(1/p(1,n)):
seq(a(n), n=1..30);  # Alois P. Heinz, Sep 02 2017

a(n) = A225435(n) + A225436(n).

A225436(n)/a(n) = 1/(1 + 1/(1 + 2/(1 + ... (1 + n)))) = A000932(n)/A000085(n+1).

cp's OEIS Frontend

A111129 Decimal expansion of the continued fraction 1+1/(1+2/(1+3/(1+4/(1+5/(1+...))))).

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

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A289491 a(n) = denominator of 1/(1 + 1/(1 + 2/(1 + ... (1 + n)))).

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