A191642 Kochański's (or Kochanski's) sequence.

15, 4697, 5548, 14774, 33696, 61072, 111231, 115985, 173819, 563316, 606004, 1751458, 1952544, 3046715, 4397195, 45051907, 653475595, 734915444, 1241384578, 2438767174, 2557084119, 5090226634, 6088149715, 18483120028, 44254634530, 48502484589, 70835215004

Offset: 1

Author

Henryk Fuks, Jun 09 2011

Keywords

• nonn

Comments

The sequence of "genitores" used to generate approximants of Pi.

References

• A. A. Kochański, Observationes cyclometricae ad facilitandam praxin accomodatae, Acta Eruditorum 4 (1685) 394-398.

Links

• Klaus Brockhaus, Table of n, a(n) for n = 1..1000
• 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.
• Henryk Fukś, Magic Squares of Subtraction of Adam Adamandy Kochański, in Research in History and Philosophy of Mathematics, Proceedings of the Canadian Society for History and Philosophy of Mathematics (CSHPM), 2017, pp. 81-95.
• Adam Adamany Kochański, Observationes Cyclometricae ad facilitandam Praxin accomodatae, original Latin text from Acta Eruditorum 4, 394-396 (1685), with English translation and annotations (by Henryk Fuks); arXiv:1106.1808 [math.HO], 2011.
• Wikipedia, Adam Adamandy Kochański

Crossrefs

Cf. A199657, A199658, A199671, A199672.

Programs

Maple

Digits := 100;
alpha:=Pi;
a:= floor(alpha);
g:=(R,S)->floor( (alpha-a)/(R-alpha*S));
S[1]:=floor(1/(alpha-a));
R[1]:=1+a*S[1];
for n from 2 to 10 do
S[n] := S[n-1]*(g(R[n-1], S[n-1])+1)+1:
R[n] := R[n-1]*(g(R[n-1], S[n-1])+1)+a:
end do:
seq(g(R[i], S[i]), i = 1 .. 10);

Mathematica

g[x_, y_] = Floor[N[(Pi - 3)/(x - Pi*y), 200]];
R = 22; S = 7;
Reap[For[i = 1, i <= 27, i++, b = g[R, S]; S = S*(b+1)+1; R = R*(b+1)+3; Print[b]; Sow[b]]][[2, 1]]; (* Jean-François Alcover, Feb 21 2019, from PARI *)

PARI

default(realprecision, 1000);
g(x,y)=floor( (Pi-3)/(x-Pi*y))
R=22; S=7; for(i=1,35, b=g(R,S); S=S*(b+1)+1; R=R*(b+1)+3; print1(b,", "))

Extensions

I added the unaccented version of the name to the definition, to make it easier to search for. - N. J. A. Sloane, Jan 12 2012