The OEIS is supported by the many generous donors to the OEIS Foundation.
Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.
Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.
%I #30 Sep 16 2021 12:20:15
%S 0,4,3,19,160,1744,644,2529,183296,3763456,4317632,54743776,
%T 1013549056,30594128896,35618973952,10392576224,3111643512832,
%U 123968232030208,48501417558016,1083228572868608,4080033616887808,188557135970304,3781715948011520
%N Numerator of 4/(1 + 1^2/(3 + 2^2/(5 + 3^2/(7 + ... + (n-1)^2/(2*n-1) )))).
%C The limit of a(n)/A345259(n) is Pi.
%H Seiichi Manyama, <a href="/A345125/b345125.txt">Table of n, a(n) for n = 0..1000</a>
%H Frits Beukers, <a href="http://www.nieuwarchief.nl/serie5/pdf/naw5-2000-01-4-372.pdf">A rational approach to Pi</a>, Nieuw Archief voor de Wiskunde, December 2000, pp. 372-379.
%F a(n)/A345259(n) = 4 * A054765(n)/A012244(n).
%e 4/(1 + 1^2/(3 + 2^2/5)) = 19/6. So a(3) = 19.
%e 0, 4, 3, 19/6, 160/51, 1744/555, 644/205, 2529/805, 183296/58345, ...
%t nmax = 25; Join[{0}, Table[4/(1 + ContinuedFractionK[j^2, (2*j + 1), {j, 1, k}]), {k, 0, nmax}] // Numerator] (* _Vaclav Kotesovec_, Sep 16 2021 *)
%o (PARI) a(n) = my(x=0); forstep(i=n, 2, -1, x = (i-1)^2/((2*i-1)+x);); if (n, numerator(4/(1+x)), numerator(x)); \\ _Michel Marcus_, Sep 16 2021
%Y Cf. A012244, A054765, A054766, A345259 (denominator).
%K nonn,frac
%O 0,2
%A _Seiichi Manyama_, Sep 16 2021