A199658 Denominators of lower 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.

8, 106, 530762, 2945294501, 43521624105025, 1466603908374792097, 89571092024800092397857, 9963245273671152951934207006, 1155597392139966274078899403965586, 200866514921276434616104042029044754594, 113151972691506812691685713772827327500605957, 68570669785555705551463950663318228291679702401993

Offset

1,1

Comments

The corresponding numerators are given in A199657.

See A199657 for more information and references.

Links

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

Henryk Fukś, Observationes Cyclometricae by Adam Adamandy Kochański - Latin text with annotated English translation, arXiv:1106.1808 [math.HO] 9 Jun 2011.

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) = 8; S(1) = A199672(1) = 7;

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

S(n) = S(n-1)*(A191642(n-1) + 1) + 1;

Example

a(1) = 8 because Kochański's first lower bound was 25/8 = A199657(1)/a(1) and his first upper bound was 22/7 = A199671(1)/A199672(1).
a(2) = S(1) * A191642(1) + 1 = 7*15 + 1 = 105 + 1 = 106,
S(2) = S(1) * (A191642(1) + 1 ) + 1 = 7*(15 + 1) + 1 = 113 = A199672(2).

Crossrefs

Cf. A191642, A199657, A199671, A199672.

Sequence in context: A236953 A345474 A000845 * A027013 A365818 A239985

Adjacent sequences: A199655 A199656 A199657 * A199659 A199660 A199661

Keyword

nonn,frac

Author

Jonathan Vos Post, Nov 08 2011

Extensions

More terms from Hugo Pfoertner, Mar 07 2020

Status

approved