%I #27 Mar 18 2022 13:06:45

%S 1,2,3,3,6,17,23,25,27,73,84,201,750,24981,46882,119318,121154,242807,

%T 276226,3009377,3383197,37871208,45930966,261728403,281868388,

%U 3021299588,3230725884,13646315477,30951797814,80602040381,1016719946612,49448385811777

%N Engel expansion of sqrt(3).

%H T. D. Noe, <a href="/A028257/b028257.txt">Table of n, a(n) for n = 1..300</a>

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

%H <a href="/index/Ed#Egypt">Index entries for sequences related to Egyptian fractions</a>

%F 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.

%t 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 *)

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

%K nonn

%O 1,2

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

%E Better name and more terms from _Simon Plouffe_

%E More terms from _Sean A. Irvine_, Dec 16 2019