A113013 Denominators of convergents to 1 + 2/(3 + 4/(5 + 6/(7 + ...))).

1, 3, 19, 151, 1511, 18131, 50767, 4061359, 73104463, 1462089259, 32165963699, 30879325151, 20071561348151, 562003717748227, 16860111532446811, 539523569038297951, 3668760269460426067, 660376848502876692059, 25094320243109314298243, 1003772809724372571929719, 42158458008423648021048199

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..400

Eric Weisstein's World of Mathematics, Continued Fraction Constants

Eric Weisstein's World of Mathematics, Generalized Continued Fraction

Formula

a(n) = denominators of 1/((sqrt(e) * Gamma(n+2))/Gamma(n+2, -1/2) - 1), where Gamma(x, a) is the incomplete Gamma function. - Eric W. Weisstein, Apr 14 2018

Example

1, 5/3, 29/19, 233/151, 2329/1511, ...

Mathematica

f[n_] := Fold[ Last@ #2 + First@ #2/#1 &, 2n - 1, Partition[ Reverse@ Range[ 2n - 2], 2]]; Denominator[ Array[ f, 18]] (* Robert G. Wilson v, Jul 07 2012 *)
(* It is interesting to note that FoldList[2 #1*#2 - (-1)^#2 &, 0, Range[19]] matches many of the terms. - Robert G. Wilson v, Jul 07 2012 *)
a[ n_] := If[ n < 0, 0, Denominator[ 1 + ContinuedFractionK[2 i, 2 i + 1, {i, 1, n}]]]; (* Michael Somos, Apr 14 2018 *)
Table[1 + ContinuedFractionK[2 k, 2 k + 1, {k, n}], {n, 0, 20}] // Denominator (* Eric W. Weisstein, Apr 14 2018 *)
Table[1/((Sqrt[E] Gamma[n + 2])/Gamma[n + 2, -1/2] - 1), {n, 0, 20}] // Denominator (* Eric W. Weisstein, Apr 14 2018 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = contfracpnqn( matrix(2, n, j, i, [2*i, 2*i+1] [j]) ); denominator( 1 + A[2, 1] / A[1, 1]) )}; /* Michael Somos, Apr 14 2018 */

Crossrefs

Cf. A113011, A113012.

Sequence in context: A007112 A007111 A346385 * A307489 A323668 A235134

Adjacent sequences: A113010 A113011 A113012 * A113014 A113015 A113016

Keyword

nonn,frac

Author

Eric W. Weisstein, Oct 10 2005

Status

approved