A028257 Engel expansion of sqrt(3).

1, 2, 3, 3, 6, 17, 23, 25, 27, 73, 84, 201, 750, 24981, 46882, 119318, 121154, 242807, 276226, 3009377, 3383197, 37871208, 45930966, 261728403, 281868388, 3021299588, 3230725884, 13646315477, 30951797814, 80602040381, 1016719946612, 49448385811777

Offset: 1

Author

Naoki Sato (naoki(AT)math.toronto.edu)

Keywords

• nonn

Links

• T. D. Noe, Table of n, a(n) for n = 1..300
• Eric Weisstein's World of Mathematics, Engel Expansion
• Index entries for sequences related to Egyptian fractions

Crossrefs

Cf. A006784 (for definition of Engel expansion), A220335.

Programs

Mathematica

EngelExp[A_,n_]:=Join[Array[1&,Floor[A]],First@Transpose@NestList[{Ceiling[1/Expand[ #[[1]]#[[2]]-1]],Expand[ #[[1]]#[[2]]-1]}&,{Ceiling[1/(A-Floor[A])],A-Floor[A]},n-1]]; EngelExp[N[3^(1/2),7! ],50] (* Vladimir Joseph Stephan Orlovsky, Jun 08 2009 *)

Formula

For a number x (here sqrt(3)), define a(1) <= a(2) <= a(3) <= ... so that x = 1/a(1) + 1/(a(1)*a(2)) + 1/(a(1)*a(2)*a(3)) + ... by x(1) = x, a(n) = ceiling(1/x(n)), x(n+1) = x(n)*a(n) - 1.

Extensions

Better name and more terms from Simon Plouffe

More terms from Sean A. Irvine, Dec 16 2019