A345259 Denominator of 4/(1 + 1^2/(3 + 2^2/(5 + 3^2/(7 + ... + (n-1)^2/(2*n-1) )))).

1, 1, 1, 6, 51, 555, 205, 805, 58345, 1197945, 1374345, 17425485, 322622685, 9738413685, 11337871545, 3308059755, 990466892415, 39460313827935, 15438480702645, 344802363740835, 1298715036217599, 60019600489849, 1203757572990973

Offset: 0

Author

Seiichi Manyama, Sep 16 2021

Keywords

• nonn
• frac

Comments

The limit of A345125(n)/a(n) is Pi.

Examples

			4/(1 + 1^2/(3 + 2^2/5)) = 19/6. So a(3) = 6.
0, 4, 3, 19/6, 160/51, 1744/555, 644/205, 2529/805, 183296/58345, ...
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000
• Frits Beukers, A rational approach to Pi, Nieuw Archief voor de Wiskunde, December 2000, pp. 372-379.

Crossrefs

Cf. A012244, A054765, A054766, A345125 (numerator).

Programs

Mathematica

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 *)

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

Formula

A345125(n)/a(n) = 4 * A054765(n)/A012244(n).