OFFSET
1,2
LINKS
T. D. Noe, Table of n, a(n) for n = 1..300
Eric Weisstein's World of Mathematics, Engel Expansion
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
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