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

%I A191642 #52 Mar 08 2020 14:45:22
%S A191642 15,4697,5548,14774,33696,61072,111231,115985,173819,563316,606004,
%T A191642 1751458,1952544,3046715,4397195,45051907,653475595,734915444,
%U A191642 1241384578,2438767174,2557084119,5090226634,6088149715,18483120028,44254634530,48502484589,70835215004
%N A191642 Kochański's (or Kochanski's) sequence.
%C A191642 The sequence of "genitores" used to generate approximants of Pi.
%D A191642 A. A. Kochański, Observationes cyclometricae ad facilitandam praxin accomodatae, Acta Eruditorum 4 (1685) 394-398.
%H A191642 Klaus Brockhaus, <a href="/A191642/b191642.txt">Table of n, a(n) for n = 1..1000</a>
%H A191642 Henryk Fuks, <a href="http://arxiv.org/abs/1111.1739">Adam Adamandy Kochanski's approximations of pi: reconstruction of the algorithm</a>, arXiv preprint arXiv:1111.1739 [math.HO], 2011-2014; Math. Intelligencer, Vol. 34 (No. 4), 2012, pp. 40-45.
%H A191642 Henryk Fukś, <a href="https://doi.org/10.1007/978-3-319-64551-3_6">Magic Squares of Subtraction of Adam Adamandy Kochański</a>, in Research in History and Philosophy of Mathematics, Proceedings of the Canadian Society for History and Philosophy of Mathematics (CSHPM), 2017, pp. 81-95.
%H A191642 Adam Adamany Kochański, <a href="http://arxiv.org/abs/1106.1808">Observationes Cyclometricae ad facilitandam Praxin accomodatae</a>, original Latin text from Acta Eruditorum 4, 394-396 (1685), with English translation and annotations (by Henryk Fuks); arXiv:1106.1808 [math.HO], 2011.
%H A191642 Wikipedia, <a href="http://en.wikipedia.org/wiki/Adam_Adamandy_Kochański">Adam Adamandy Kochański</a>
%p A191642 Digits := 100;
%p A191642 alpha:=Pi;
%p A191642 a:= floor(alpha);
%p A191642 g:=(R,S)->floor( (alpha-a)/(R-alpha*S));
%p A191642 S[1]:=floor(1/(alpha-a));
%p A191642 R[1]:=1+a*S[1];
%p A191642 for n from 2 to 10 do
%p A191642 S[n] := S[n-1]*(g(R[n-1], S[n-1])+1)+1:
%p A191642 R[n] := R[n-1]*(g(R[n-1], S[n-1])+1)+a:
%p A191642 end do:
%p A191642 seq(g(R[i], S[i]), i = 1 .. 10);
%t A191642 g[x_, y_] = Floor[N[(Pi - 3)/(x - Pi*y), 200]];
%t A191642 R = 22; S = 7;
%t A191642 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 *)
%o A191642 (PARI)
%o A191642 default(realprecision, 1000);
%o A191642 g(x,y)=floor( (Pi-3)/(x-Pi*y))
%o A191642 R=22; S=7; for(i=1,35, b=g(R,S); S=S*(b+1)+1; R=R*(b+1)+3; print1(b,", "))
%Y A191642 Cf. A199657, A199658, A199671, A199672.
%K A191642 nonn
%O A191642 1,1
%A A191642 _Henryk Fuks_, Jun 09 2011
%E A191642 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