A113012 Numerators of convergents to 1 + 2/(3 + 4/(5 + 6/(7 + ...))).

1, 5, 29, 233, 2329, 27949, 78257, 6260561, 112690097, 2253801941, 49583642701, 47600296993, 30940193045449, 866325405272573, 25989762158177189, 831672389061670049, 5655372245619356333, 1017967004211484139941, 38682746160036397317757, 1547309846401455892710281

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) = Numerator(Sum_{k=0..n+1} (-1)^k*1/(2k)!!). - Alexander Adamchuk, Jul 02 2006

a(n) <= A000354(n+1). - Michael Somos, Sep 28 2017

Example

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

Mathematica

Numerator[Table[Sum[(-1)^k*1/(2k)!!, {k, 0, n}], {n, 1, 25}]] (* Alexander Adamchuk, Jul 02 2006 *)
f[n_] := Fold[ Last@ #2 + First@ #2/#1 &, 2n - 1, Partition[ Reverse@ Range[ 2n - 2], 2]]; Numerator[ Array[ f, 18]]  (* Robert G. Wilson v, Jul 07 2012 *)
a[ n_] := If[ n < 0, 0, Numerator[ 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}] // Numerator (* Eric W. Weisstein, Apr 14 2018 *)
Table[1/((Sqrt[E] Gamma[n + 2])/Gamma[n + 2, -1/2] - 1), {n, 0, 20}] // Numerator (* 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]) ); numerator( 1 + A[2, 1] / A[1, 1]) )}; /* Michael Somos, Apr 14 2018 */
(GAP) List(List([0..20], n->Sum([0..n], k->(-1)^k*(1/(Product([0..Int(2*k/2)-1], i->2*k-2*i))))), NumeratorRat); # Muniru A Asiru, Apr 14 2018

Crossrefs

Cf. A000354, A113011, A113013.

Sequence in context: A182018 A309260 A087662 * A000354 A103815 A134752

Adjacent sequences: A113009 A113010 A113011 * A113013 A113014 A113015

Keyword

nonn,frac

Author

Eric W. Weisstein, Oct 10 2005

Status

approved