A154956 Pierce expansion of 2/Pi.

1, 2, 3, 5, 10, 71, 868, 1788, 7455, 44266, 54626, 74153, 224166, 390471, 1489304, 3737961, 22277163, 37201631, 113275744, 165029426, 2642368758, 3362202939, 5191046363, 8438525012, 36226438506, 40174126779, 125336047846, 531802867080, 599020778171

Offset: 0

Author

Jaume Oliver Lafont, Jan 18 2009

Keywords

• nonn

Examples

			1 - 1/2(1 - 1/3(1 - 1/5(1 - 1/10(1 - 1/71)))) = 2/(355/113).
		

Links

• Simon Plouffe and G. C. Greubel, Table of n, a(n) for n = 0..500 (terms from 0 to 216 were computed by Simon Plouffe)

Crossrefs

Cf. A006283 (1/Pi), A061233 (4 - Pi).

Cf. A060294 (decimal expansion of 2/Pi). - R. J. Mathar, Jan 21 2009

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: a060294 := evalf(2/Pi) ; Pierce(a060294) ; # R. J. Mathar, Jan 21 2009

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[2/Pi, 7!], 50] (* G. C. Greubel, Nov 13 2016 *)

PARI

A154956(N=99)={localprec(N); my(c=2/Pi, d=c+c/10^N, a=[1\c]); while(a[#a]==1\d&&c=1-c*a[#a], d=1-d*a[#a]; a=concat(a, 1\c)); a[^-1]}  \\ The optional argument is the precision used, approx. equal to the total number of digits in the result. - M. F. Hasler, Jul 04 2016

Extensions

More terms from R. J. Mathar, Jan 21 2009

Offset in b-file corrected by N. J. A. Sloane, Aug 31 2009