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

1,2

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

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.

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 *)

Crossrefs

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

Sequence in context: A329748 A173094 A087989 * A100228 A111003 A347904

Adjacent sequences: A028254 A028255 A028256 * A028258 A028259 A028260

Keyword

nonn

Author

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

Extensions

Better name and more terms from Simon Plouffe

More terms from Sean A. Irvine, Dec 16 2019

Status

approved