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

%I A113012 #55 Feb 16 2025 08:32:59
%S A113012 1,5,29,233,2329,27949,78257,6260561,112690097,2253801941,49583642701,
%T A113012 47600296993,30940193045449,866325405272573,25989762158177189,
%U A113012 831672389061670049,5655372245619356333,1017967004211484139941,38682746160036397317757,1547309846401455892710281
%N A113012 Numerators of convergents to 1 + 2/(3 + 4/(5 + 6/(7 + ...))).
%H A113012 G. C. Greubel, <a href="/A113012/b113012.txt">Table of n, a(n) for n = 0..400</a>
%H A113012 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ContinuedFractionConstants.html">Continued Fraction Constants</a>
%H A113012 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GeneralizedContinuedFraction.html">Generalized Continued Fraction</a>
%F A113012 a(n) = Numerator(Sum_{k=0..n+1} (-1)^k*1/(2k)!!). - _Alexander Adamchuk_, Jul 02 2006
%F A113012 a(n) <= A000354(n+1). - _Michael Somos_, Sep 28 2017
%e A113012 1, 5/3, 29/19, 233/151, 2329/1511, ...
%t A113012 Numerator[Table[Sum[(-1)^k*1/(2k)!!,{k,0,n}],{n,1,25}]] (* _Alexander Adamchuk_, Jul 02 2006 *)
%t A113012 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 *)
%t A113012 a[ n_] := If[ n < 0, 0, Numerator[ 1 + ContinuedFractionK[2 i, 2 i + 1, {i, 1, n}]]]; (* _Michael Somos_, Apr 14 2018 *)
%t A113012 Table[1 + ContinuedFractionK[2 k, 2 k + 1, {k, n}], {n, 0, 20}] // Numerator (* _Eric W. Weisstein_, Apr 14 2018 *)
%t A113012 Table[1/((Sqrt[E] Gamma[n + 2])/Gamma[n + 2, -1/2] - 1), {n, 0, 20}] // Numerator (* _Eric W. Weisstein_, Apr 14 2018 *)
%o A113012 (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 */
%o A113012 (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
%Y A113012 Cf. A000354, A113011, A113013.
%K A113012 nonn,frac
%O A113012 0,2
%A A113012 _Eric W. Weisstein_, Oct 10 2005