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.

22, 355, 1667793, 9254583360, 136736469144003, 4607608800745469094, 281400492287928033977865, 31300739558170811075425879683, 3630447578393999693394346080912441, 631044398076445026705805235784200494623, 355478037191228783108834088006248470029544494, 215421467928070598707869001502226604080254111086473

Offset: 1

Author

Jonathan Vos Post, Nov 08 2011

Keywords

• nonn
• frac

Comments

The corresponding denominators are given in A199672.

See A199657 for more information and references.

Examples

			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).
a(2) = a(1) * (A191642(1) + 1) + 3 = 22*(15 + 1) + 3 = 352 + 3 = 355,
a(3) = a(2) * (A191642(2) + 1) + 3 = 355*(4697 + 1) + 3 = 1667793,
a(4) = a(3) * (A191642(3) + 1) + 3 = 1667793*(5548 + 1) + 3 = 9254583360.
		

Links

• 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.

Crossrefs

Cf. A191642, A199657, A199658, A199672.

Programs

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 *)

Formula

a(1) = 22;

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

Extensions

More terms from Hugo Pfoertner, Mar 07 2020