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

1,1

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

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 *)

Prog

(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, ", "))

Crossrefs

Cf. A199657, A199658, A199671, A199672.

Sequence in context: A027513 A287037 A229931 * A348816 A206387 A198903

Adjacent sequences: A191639 A191640 A191641 * A191643 A191644 A191645

Keyword

nonn

Author

Henryk Fuks, Jun 09 2011

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

Status

approved