

A199671


Numerators of upper rational approximants of pi, given by Adam Adamandy Kochanski in 1685.


2




OFFSET

1,1


COMMENTS

A199672 has denominators of upper rational approximants of pi, given by Adam Adamandy Kochanski in 1685. In his 1685 paper "Observationes cyclometricae" published in Acta Eruditorum, Adam Adamandy Kochanski presented an approximate rulerandcompass construction for rectification of the circle. It is not generally known that the first part of this paper included an interesting sequence of rational approximations of pi.


LINKS

Table of n, a(n) for n=1..5.
Henryk Fuks, Adam Adamandy Kochanski's approximations of pi: reconstruction of the algorithm, arXiv preprint arXiv:1111.1739 [math.HO], 20112014. Math. Intelligencer, Vol. 34 (No. 4), 2012, pp. 4045.


EXAMPLE

a(1) = 22 because his first lower bound was 25/8 and his first upper bound was 22/7.


MATHEMATICA

g[x_, y_] = Floor[N[(Pi  3)/(x  Pi*y), 200]];
R = 22; S = 7;
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]] (* JeanFrançois Alcover, Feb 21 2019 *)


CROSSREFS

Cf. A199657, A199658, A199671.
Sequence in context: A208458 A016263 A001718 * A253878 A081127 A016196
Adjacent sequences: A199668 A199669 A199670 * A199672 A199673 A199674


KEYWORD

nonn,frac


AUTHOR

Jonathan Vos Post, Nov 08 2011


STATUS

approved



