A199672 Denominators of upper rational approximants of Pi with the first 5 terms given by Adam Adamandy Kochański in 1685, continued using a reconstruction by Fukś that is highly likely to match Kochański's incompletely published method.

7, 113, 530875, 2945825376, 43524569930401, 1466647432944722498, 89572558672233037120355, 9963334846229825184971327361, 1155607355474812503904084375292947, 200867670528631909428607946113420047541, 113152173559177341323595142380773440920653498, 68570782937729264728805274258460609065120623055491

Offset: 1

Author

Jonathan Vos Post, Nov 08 2011

Keywords

• nonn
• frac

Comments

The corresponding numerators are given in A199671.

See A199657 for more information and references.

Examples

			a(1) = 7 because Kochański's first lower bound was 25/8 = A199657(1)/A199658(1) and his first upper bound was 22/7 = A199671(1)/a(1).
a(2) = a(1) * (A191642(1) + 1) + 1 = 7*(15 + 1) + 1 = 112 + 1 = 113,
a(3) = a(2) * (A191642(2) + 1) + 1 = 113*(4697 + 1) + 1 = 530875,
a(4) = a(3) * (A191642(3) + 1) + 1 = 530875*(5548 + 1) + 1 = 2945825376.
		

Links

• Henryk Fukś, Adam Adamandy Kochański's approximations of pi: reconstruction of the algorithm, arXiv preprint arXiv:1111.1739 [math.HO], 2011. Math. Intelligencer, Vol. 34 (No. 4), 2012, pp. 40-45.

Crossrefs

Cf. A191642, A199657, A199658, A199671 (numerators).

Formula

a(1) = 7;

a(n) = a(n-1)*(A191642(n-1) + 1) + 1, where A191642 are Kochański's "genitores".

Extensions

More terms from Hugo Pfoertner, Mar 07 2020