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%I #19 Mar 05 2023 13:25:34
%S 1,11,13,59,582,12285,127893,654577,1896651,2083263,3828867,6195679,
%T 22339606,43877386,209882043,269091773,1585394894,2614512078,
%U 3726537414,4487682121,6296491774,8648456991,23933983277,174313954158,367633382556
%N Pierce expansion of Catalan's Constant A006752.
%H G. C. Greubel, <a href="/A132201/b132201.txt">Table of n, a(n) for n = 1..1000</a>
%H T. A. Pierce, <a href="http://www.jstor.org/stable/2299963">On an algorithm and its use in approximating roots of algebraic equations</a>, Am. Math. Monthly 36 (10) (1929) 523.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PierceExpansion.html">Pierce Expansion</a>.
%e 0.9159... = 1/1 - 1/11 + 1/(11*13) - 1/(11*13*59) + 1/(11*13*59*582) - ...
%p Digits := 300: Pierce := proc(x) local resid,a,i,an ; resid := x ; a := [] ; for i from 1 do an := floor(1./resid) ; a := [op(a),an] ; resid := evalf(1.-an*resid) ; if ilog10( mul(i,i=a)) > 0.7*Digits then break ; fi ; od: RETURN(a) ; end: Pierce(Catalan);
%t PierceExp[A_, n_] := Join[Array[1 &, Floor[A]], First@Transpose@ NestList[{Floor[1/Expand[1 - #[[1]] #[[2]]]], Expand[1 - #[[1]] #[[2]]]} &, {Floor[1/(A - Floor[A])], A - Floor[A]}, n - 1]]; PierceExp[N[Catalan , 7!], 20] (* _G. C. Greubel_, Nov 15 2016 *)
%o (PARI) r=1/Catalan; for(n=1, 10, print(floor(r), ", "); r=r/(r-floor(r))) \\ _G. C. Greubel_, Nov 15 2016
%Y Cf. A006752.
%K nonn
%O 1,2
%A _R. J. Mathar_, Nov 05 2007
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