A140076 Pierce expansion of the cube root of 1/2.

1, 4, 5, 7, 8, 18, 384, 7958, 14304, 16623, 18610, 20685, 72923, 883177, 1516692, 2493788, 2504069, 22881179, 110219466, 2241255405, 34982468090, 64356019489, 110512265214, 1142808349967, 3550630472116, 5238523454726, 7129035664265

Offset

1,2

Comments

2^(-1/3) = 1-1/4(1-1/5(1-1/7(1-1/8(1-1/18(1-1/384(...))))))

Links

G. C. Greubel, Table of n, a(n) for n = 1..500

G. P. Michon, Pierce Expansions.

Eric Weisstein's World of Mathematics, Pierce Expansion.

Formula

Starting with x(1)=2^(-1/3), a(n) = floor(1/x(n)) and x(n+1) = 1-a(n)x(n).

Example

a(1) is 1 because the floor of 2^(1/3) is 1.
a(2)=4 because 1/(1-2^(-1/3)) is 4.8473221...

Mathematica

$MaxExtraPrecision = 80; x[1] = 2^(-1/3); a[n_] := a[n] = Floor[1/x[n]]; x[n_] := x[n] = 1 - a[n-1]*x[n-1]; Table[a[n], {n, 1, 27}] (* Jean-François Alcover, Dec 12 2011 *)
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^(-1/3), 7!], 25] (* G. C. Greubel, Nov 14 2016 *)

Crossrefs

Cf. A091831, A006283, A006284, A061233, A118242.

Sequence in context: A276324 A226628 A335842 * A135186 A011336 A094328

Adjacent sequences: A140073 A140074 A140075 * A140077 A140078 A140079

Keyword

easy,nice,nonn

Author

Gerard P. Michon, Jun 01 2008

Status

approved