A132201 Pierce expansion of Catalan's Constant A006752.

1, 11, 13, 59, 582, 12285, 127893, 654577, 1896651, 2083263, 3828867, 6195679, 22339606, 43877386, 209882043, 269091773, 1585394894, 2614512078, 3726537414, 4487682121, 6296491774, 8648456991, 23933983277, 174313954158, 367633382556

Offset: 1

Author

R. J. Mathar, Nov 05 2007

Keywords

• nonn

Examples

			0.9159... = 1/1 - 1/11 + 1/(11*13) - 1/(11*13*59) + 1/(11*13*59*582) - ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• T. A. Pierce, On an algorithm and its use in approximating roots of algebraic equations, Am. Math. Monthly 36 (10) (1929) 523.
• Eric Weisstein's World of Mathematics, Pierce Expansion.

Crossrefs

Cf. A006752.

Programs

Maple

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);

Mathematica

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 *)

PARI

r=1/Catalan; for(n=1, 10, print(floor(r), ", "); r=r/(r-floor(r))) \\ G. C. Greubel, Nov 15 2016