cp's OEIS Frontend

A199671 Numerators 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 A199671 #33 Mar 13 2020 06:16:32
%S A199671 22,355,1667793,9254583360,136736469144003,4607608800745469094,
%T A199671 281400492287928033977865,31300739558170811075425879683,
%U A199671 3630447578393999693394346080912441,631044398076445026705805235784200494623,355478037191228783108834088006248470029544494,215421467928070598707869001502226604080254111086473
%N A199671 Numerators 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 A199671 The corresponding denominators are given in A199672.
%C A199671 See A199657 for more information and references.
%H A199671 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 A199671 a(1) = 22;
%F A199671 a(n) = a(n-1)*(A191642(n-1) + 1) + 3, where A191642 are Kochański's "genitores".
%e A199671 a(1) = 22 because Kochański's first lower bound was 25/8 = A199657/A199658(1) and his first upper bound was 22/7 = a(1)/A199672(1).
%e A199671 a(2) = a(1) * (A191642(1) + 1) + 3 = 22*(15 + 1) + 3 = 352 + 3 = 355,
%e A199671 a(3) = a(2) * (A191642(2) + 1) + 3 = 355*(4697 + 1) + 3 = 1667793,
%e A199671 a(4) = a(3) * (A191642(3) + 1) + 3 = 1667793*(5548 + 1) + 3 = 9254583360.
%t A199671 g[x_, y_] = Floor[N[(Pi - 3)/(x - Pi*y), 200]];
%t A199671 R = 22; S = 7;
%t A199671 Reap[Print[R]; Sow[R]; For[i = 1, i <= 4, i++, b = g[R, S]; S = S*(b+1)+1; R = R*(b+1)+3; Print[R]; Sow[R]]][[2, 1]] (* _Jean-François Alcover_, Feb 21 2019 *)
%Y A199671 Cf. A191642, A199657, A199658, A199672.
%K A199671 nonn,frac
%O A199671 1,1
%A A199671 _Jonathan Vos Post_, Nov 08 2011
%E A199671 More terms from _Hugo Pfoertner_, Mar 07 2020