cp's OEIS Frontend

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.

%I A199658 #37 Mar 13 2020 08:04:43
%S A199658 8,106,530762,2945294501,43521624105025,1466603908374792097,
%T A199658 89571092024800092397857,9963245273671152951934207006,
%U A199658 1155597392139966274078899403965586,200866514921276434616104042029044754594,113151972691506812691685713772827327500605957,68570669785555705551463950663318228291679702401993
%N 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.
%C A199658 The corresponding numerators are given in A199657.
%C A199658 See A199657 for more information and references.
%H A199658 Henryk Fukś, <a href="https://arxiv.org/abs/1106.1808">Observationes Cyclometricae by Adam Adamandy Kochański - Latin text with annotated English translation</a>, arXiv:1106.1808 [math.HO] 9 Jun 2011.
%H A199658 Henryk Fukś, <a href="http://arxiv.org/abs/1111.1739">Adam Adamandy Kochański's approximations of pi: reconstruction of the algorithm</a>, arXiv preprint arXiv:1111.1739 [math.HO], 2011. Math. Intelligencer, Vol. 34 (No. 4), 2012, pp. 40-45.
%F A199658 a(1) = 8; S(1) = A199672(1) = 7;
%F A199658 a(n) = S(n-1)*A191642(n-1) + 1, where A191642 are Kochański's "genitores";
%F A199658 S(n) = S(n-1)*(A191642(n-1) + 1) + 1;
%e A199658 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).
%e A199658 a(2) = S(1) * A191642(1) + 1 = 7*15 + 1 = 105 + 1 = 106,
%e A199658 S(2) = S(1) * (A191642(1) + 1 ) + 1 = 7*(15 + 1) + 1 = 113 = A199672(2).
%Y A199658 Cf. A191642, A199657, A199671, A199672.
%K A199658 nonn,frac
%O A199658 1,1
%A A199658 _Jonathan Vos Post_, Nov 08 2011
%E A199658 More terms from _Hugo Pfoertner_, Mar 07 2020