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
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