OFFSET
1,2
COMMENTS
For a number x (here sqrt(2)), 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.
LINKS
T. D. Noe, Table of n, a(n) for n = 1..300
Benoît Rittaud, La porte d’harmonie, Images des Mathématiques, CNRS, 2009 (in French).
Naoki Sato, Home page (broken link)
Eric Weisstein's World of Mathematics, Engel Expansion
Eric Weisstein's World of Mathematics, Pythagoras's Constant
EXAMPLE
sqrt(2) = 1.4142135623730950488...
1 + 1/3 = 4/3 = 1.3333333333333333333...; sqrt(2) - 4/3 = 0.080880229...
1 + 1/3 + 1/15 = 7/5 = 1.4; sqrt(2) - 7/5 = 0.014213562373...
1 + 1/3 + 1/15 + 1/75 = 106/75 = 1.4133333333333333...; sqrt(2) - 106/75 = 0.000880229...
MATHEMATICA
expandEngel[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]]; expandEngel[N[2^(1/2), 7!], 47] (* Vladimir Joseph Stephan Orlovsky, Jun 08 2009 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Naoki Sato (naoki(AT)math.toronto.edu)
EXTENSIONS
More terms from Simon Plouffe, Jan 05 2002
STATUS
approved