cp's OEIS Frontend

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.

%I A199672 #37 Mar 13 2020 04:29:24
%S A199672 7,113,530875,2945825376,43524569930401,1466647432944722498,
%T A199672 89572558672233037120355,9963334846229825184971327361,
%U A199672 1155607355474812503904084375292947,200867670528631909428607946113420047541,113152173559177341323595142380773440920653498,68570782937729264728805274258460609065120623055491
%N 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.
%C A199672 The corresponding numerators are given in A199671.
%C A199672 See A199657 for more information and references.
%H A199672 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 A199672 a(1) = 7;
%F A199672 a(n) = a(n-1)*(A191642(n-1) + 1) + 1, where A191642 are Kochański's "genitores".
%e A199672 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).
%e A199672 a(2) = a(1) * (A191642(1) + 1) + 1 = 7*(15 + 1) + 1 = 112 + 1 = 113,
%e A199672 a(3) = a(2) * (A191642(2) + 1) + 1 = 113*(4697 + 1) + 1 = 530875,
%e A199672 a(4) = a(3) * (A191642(3) + 1) + 1 = 530875*(5548 + 1) + 1 = 2945825376.
%Y A199672 Cf. A191642, A199657, A199658, A199671 (numerators).
%K A199672 nonn,frac
%O A199672 1,1
%A A199672 _Jonathan Vos Post_, Nov 08 2011
%E A199672 More terms from _Hugo Pfoertner_, Mar 07 2020