A006283 Pierce expansion for 1 / Pi. (Formerly M3092)

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

0,1

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

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

Example

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

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 *)

Prog

(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]

Crossrefs

Cf. A154956 (analog for 2/Pi).

Sequence in context: A352027 A061182 A143552 * A232017 A100511 A033506

Adjacent sequences: A006280 A006281 A006282 * A006284 A006285 A006286

Keyword

nonn

Author

Jeffrey Shallit

Status

approved