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

Author

Jonathan Vos Post, Nov 08 2011

Keywords

• nonn
• frac

Comments

The corresponding numerators are given in A199657.

See A199657 for more information and references.

Examples

			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).
		

Links

• 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.

Crossrefs

Cf. A191642, A199657, A199671, A199672.

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;

Extensions

More terms from Hugo Pfoertner, Mar 07 2020