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
KEYWORD
easy,nice,nonn
AUTHOR
Gerard P. Michon, Jun 01 2008
STATUS
approved