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

%I A113013 #34 Feb 16 2025 08:32:59
%S A113013 1,3,19,151,1511,18131,50767,4061359,73104463,1462089259,32165963699,
%T A113013 30879325151,20071561348151,562003717748227,16860111532446811,
%U A113013 539523569038297951,3668760269460426067,660376848502876692059,25094320243109314298243,1003772809724372571929719,42158458008423648021048199
%N A113013 Denominators of convergents to 1 + 2/(3 + 4/(5 + 6/(7 + ...))).
%H A113013 G. C. Greubel, <a href="/A113013/b113013.txt">Table of n, a(n) for n = 0..400</a>
%H A113013 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ContinuedFractionConstants.html">Continued Fraction Constants</a>
%H A113013 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GeneralizedContinuedFraction.html">Generalized Continued Fraction</a>
%F A113013 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
%e A113013 1, 5/3, 29/19, 233/151, 2329/1511, ...
%t A113013 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 *)
%t A113013 (* 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 *)
%t A113013 a[ n_] := If[ n < 0, 0, Denominator[ 1 + ContinuedFractionK[2 i, 2 i + 1, {i, 1, n}]]]; (* _Michael Somos_, Apr 14 2018 *)
%t A113013 Table[1 + ContinuedFractionK[2 k, 2 k + 1, {k, n}], {n, 0, 20}] // Denominator (* _Eric W. Weisstein_, Apr 14 2018 *)
%t A113013 Table[1/((Sqrt[E] Gamma[n + 2])/Gamma[n + 2, -1/2] - 1), {n, 0, 20}] // Denominator (* _Eric W. Weisstein_, Apr 14 2018 *)
%o A113013 (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 */
%Y A113013 Cf. A113011, A113012.
%K A113013 nonn,frac
%O A113013 0,2
%A A113013 _Eric W. Weisstein_, Oct 10 2005