This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A345259 #28 Sep 16 2021 13:31:55
%S A345259 1,1,1,6,51,555,205,805,58345,1197945,1374345,17425485,322622685,
%T A345259 9738413685,11337871545,3308059755,990466892415,39460313827935,
%U A345259 15438480702645,344802363740835,1298715036217599,60019600489849,1203757572990973
%N A345259 Denominator of 4/(1 + 1^2/(3 + 2^2/(5 + 3^2/(7 + ... + (n-1)^2/(2*n-1) )))).
%C A345259 The limit of A345125(n)/a(n) is Pi.
%H A345259 Seiichi Manyama, <a href="/A345259/b345259.txt">Table of n, a(n) for n = 0..1000</a>
%H A345259 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 A345259 A345125(n)/a(n) = 4 * A054765(n)/A012244(n).
%e A345259 4/(1 + 1^2/(3 + 2^2/5)) = 19/6. So a(3) = 6.
%e A345259 0, 4, 3, 19/6, 160/51, 1744/555, 644/205, 2529/805, 183296/58345, ...
%t A345259 nmax = 25; Join[{1}, Table[4/(1 + ContinuedFractionK[j^2, (2*j + 1), {j, 1, k}]), {k, 0, nmax}] // Denominator] (* _Vaclav Kotesovec_, Sep 16 2021 *)
%o A345259 (PARI) a(n) = my(x=0); forstep(i=n, 2, -1, x = (i-1)^2/((2*i-1)+x);); if (n, denominator(4/(1+x)), denominator(x)); \\ _Michel Marcus_, Sep 16 2021
%Y A345259 Cf. A012244, A054765, A054766, A345125 (numerator).
%K A345259 nonn,frac
%O A345259 0,4
%A A345259 _Seiichi Manyama_, Sep 16 2021