%I #33 Nov 29 2019 03:28:39
%S 1,3,5,5,16,18,78,102,120,144,251,363,1402,31169,88630,184655,259252,
%T 298770,4196070,38538874,616984563,1975413035,5345718057,27843871197,
%U 54516286513,334398528974,445879679626,495957494386,2450869042061,2629541150529,4088114099885
%N Engel expansion of sqrt(2).
%C 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.
%H T. D. Noe, <a href="/A028254/b028254.txt">Table of n, a(n) for n = 1..300</a>
%H Benoît Rittaud, <a href="http://images.math.cnrs.fr/La-porte-d-harmonie.html">La porte d’harmonie</a>, Images des Mathématiques, CNRS, 2009 (in French).
%H Naoki Sato, <a href="http://www.math.toronto.edu/~naoki/">Home page</a> (broken link)
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EngelExpansion.html">Engel Expansion</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PythagorassConstant.html">Pythagoras's Constant</a>
%e sqrt(2) = 1.4142135623730950488...
%e 1 + 1/3 = 4/3 = 1.3333333333333333333...; sqrt(2) - 4/3 = 0.080880229...
%e 1 + 1/3 + 1/15 = 7/5 = 1.4; sqrt(2) - 7/5 = 0.014213562373...
%e 1 + 1/3 + 1/15 + 1/75 = 106/75 = 1.4133333333333333...; sqrt(2) - 106/75 = 0.000880229...
%t 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 *)
%Y Cf. A002193 (decimal expansion), A006784 (for definition of Engel expansion), A028257 (Engel expansion of sqrt(3)).
%K nonn
%O 1,2
%A Naoki Sato (naoki(AT)math.toronto.edu)
%E More terms from _Simon Plouffe_, Jan 05 2002