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 A199671 Numerators of upper rational approximants of pi, given by Adam Adamandy Kochanski in 1685. 2
 22, 355, 1667793, 9254583360, 136736469144003 (list; graph; refs; listen; history; text; internal format)
 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 ruler-and-compass 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 Henryk Fuks, Adam Adamandy Kochanski's approximations of pi: reconstruction of the algorithm, arXiv preprint arXiv:1111.1739 [math.HO], 2011-2014. Math. Intelligencer, Vol. 34 (No. 4), 2012, pp. 40-45. 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]] (* Jean-François 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

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Last modified May 25 15:22 EDT 2019. Contains 323572 sequences. (Running on oeis4.)