A140076 - OEIS

%I #11 Nov 14 2016 03:56:25

%S 1,4,5,7,8,18,384,7958,14304,16623,18610,20685,72923,883177,1516692,

%T 2493788,2504069,22881179,110219466,2241255405,34982468090,

%U 64356019489,110512265214,1142808349967,3550630472116,5238523454726,7129035664265

%N Pierce expansion of the cube root of 1/2.

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

%H G. C. Greubel, <a href="/A140076/b140076.txt">Table of n, a(n) for n = 1..500</a>

%H G. P. Michon, <a href="http://www.numericana.com/answer/fractions.htm#pierce">Pierce Expansions</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PierceExpansion.html">Pierce Expansion</a>.

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

%e a(1) is 1 because the floor of 2^(1/3) is 1.

%e a(2)=4 because 1/(1-2^(-1/3)) is 4.8473221...

%t $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 *)

%t 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 *)

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

%K easy,nice,nonn

%O 1,2

%A _Gerard P. Michon_, Jun 01 2008