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

1,1

Comments

The corresponding numerators are given in A199671.

See A199657 for more information and references.

Links

Table of n, a(n) for n=1..12.

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.

Formula

a(1) = 7;

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

Example

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.

Crossrefs

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

Sequence in context: A159552 A228929 A086788 * A240288 A220343 A183403

Adjacent sequences: A199669 A199670 A199671 * A199673 A199674 A199675

Keyword

nonn,frac

Author

Jonathan Vos Post, Nov 08 2011

Extensions

More terms from Hugo Pfoertner, Mar 07 2020

Status

approved