%I #31 Mar 08 2020 14:02:13

%S 25,333,1667438,9252915567,136727214560643,4607472064276325091,

%T 281395884679127288508771,31300458157678523147391901818,

%U 3630416277654441522583270655032758,631040767628866632706111841438119582182,355477406146830706663807382201012685829049871,215421112450033407479085892668138597831784081541979

%N Numerators 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 The corresponding denominators are given in A199658.

%C The reconstruction refers to the calculation of the "genitores" in A191642, for which Kochański only announced that he would describe them in more detail in a future work: "I will explain the aforementioned method more completely in Polymathic thoughts and inventions, which work, if God prolongs my life, I have decided to put out for public benefit" (translation from Latin by H. Fukś).

%H 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 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 a(1) = 25; R(1) = A199671(1) = 22;

%F a(n) = R(n-1)*A191642(n-1) + 3, where A191642 are Kochański's "genitores";

%F R(n) = R(n-1)*(A191642(n-1) + 1) + 3;

%e a(1) = 25 because Kochański's first lower bound was 25/8 = a(1)/A199658(1) and his first upper bound was 22/7 = A199671(1)/A199672(1).

%e a(2) = R(1) * A191642(1) + 3 = 22*15 + 3 = 330 + 3 = 333,

%e R(2) = R(1) * (A191642(1) + 1 ) + 3 = 22*(15 + 1) + 3 = 355 = A199671(2).

%Y Cf. A191642, A199658, A199671, A199672.

%K nonn,frac

%O 1,1

%A _Jonathan Vos Post_, Nov 08 2011

%E More terms from _Hugo Pfoertner_, Mar 07 2020