A006283 Pierce expansion for 1 / Pi.

3, 22, 118, 383, 571, 635, 70529, 375687, 399380, 575584, 699357, 1561065, 1795712, 194445473, 253745996, 3199003690, 3727084011, 6607433185, 16248462801, 172940584814, 313728984965, 796022309187, 5348508258636, 5962546521072, 97497255361780, 121347007731845

Offset: 0

Author

Jeffrey Shallit

Keywords

• nonn

Comments

Sequence can be produced with the unit circle: a(1) = number of diameter length arcs in circle rounded down to nearest integer (remainder arc = x_1). a(2) = number of x_1 length arcs in circle rounded down to nearest integer (remainder arc = x_2). a(3) = number of x_2 length arcs in circle rounded down to nearest integer (remainder arc = x_3). And so on ... . - Peter Woodward, Sep 08 2016

Examples

			Let x_0 = 1/Pi = 0.318309886... and a(0) = floor(1/x_0) = 3. Then set x_1 = 1 - a_0*x_0 = 0.0450703..., and a(1) = floor(1/x_1) = 22. Then x_2 = 1 - a_1*x_1 = 0.008452..., and a(2) = floor(1/x2) = 118. - _Michael B. Porter_, Sep 09 2016
		

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..500
• Jeffrey Shallit, Some predictable Pierce expansions, Fib. Quart., 22 (1984), 332-335.
• Eric Weisstein's World of Mathematics, Pierce Expansion

Crossrefs

Cf. A154956 (analog for 2/Pi).

Programs

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[1/Pi, 8!], 50] (* G. C. Greubel, Nov 13 2016 *)

PARI

default(realprecision, 100000); r=Pi; for(n=1,100,s=(r/(r-floor(r))); print1(floor(r),","); r=s) \\ Benoit Cloitre [amended by Georg Fischer, Nov 20 2020]