A199657 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.

25, 333, 1667438, 9252915567, 136727214560643, 4607472064276325091, 281395884679127288508771, 31300458157678523147391901818, 3630416277654441522583270655032758, 631040767628866632706111841438119582182, 355477406146830706663807382201012685829049871, 215421112450033407479085892668138597831784081541979

Offset

1,1

Comments

The corresponding denominators are given in A199658.

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).

Links

Table of n, a(n) for n=1..12.

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.

Formula

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

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

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

Example

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).
a(2) = R(1) * A191642(1) + 3 = 22*15 + 3 = 330 + 3 = 333,
R(2) = R(1) * (A191642(1) + 1 ) + 3 = 22*(15 + 1) + 3 = 355 = A199671(2).

Crossrefs

Cf. A191642, A199658, A199671, A199672.

Sequence in context: A010977 A022589 A344656 * A094730 A265930 A265922

Adjacent sequences: A199654 A199655 A199656 * A199658 A199659 A199660

Keyword

nonn,frac

Author

Jonathan Vos Post, Nov 08 2011

Extensions

More terms from Hugo Pfoertner, Mar 07 2020

Status

approved