A345125 Numerator of 4/(1 + 1^2/(3 + 2^2/(5 + 3^2/(7 + ... + (n-1)^2/(2*n-1) )))).

0, 4, 3, 19, 160, 1744, 644, 2529, 183296, 3763456, 4317632, 54743776, 1013549056, 30594128896, 35618973952, 10392576224, 3111643512832, 123968232030208, 48501417558016, 1083228572868608, 4080033616887808, 188557135970304, 3781715948011520

Offset

0,2

Comments

The limit of a(n)/A345259(n) is Pi.

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.

Formula

a(n)/A345259(n) = 4 * A054765(n)/A012244(n).

Example

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

Mathematica

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 *)

Prog

(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

Crossrefs

Cf. A012244, A054765, A054766, A345259 (denominator).

Sequence in context: A241358 A178417 A178628 * A167479 A278424 A278662

Adjacent sequences: A345122 A345123 A345124 * A345126 A345127 A345128

Keyword

nonn,frac

Author

Seiichi Manyama, Sep 16 2021

Status

approved