A028254 Engel expansion of sqrt(2).

1, 3, 5, 5, 16, 18, 78, 102, 120, 144, 251, 363, 1402, 31169, 88630, 184655, 259252, 298770, 4196070, 38538874, 616984563, 1975413035, 5345718057, 27843871197, 54516286513, 334398528974, 445879679626, 495957494386, 2450869042061, 2629541150529, 4088114099885

Offset: 1

Author

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

Keywords

• nonn

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.

Examples

			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...
		

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

Crossrefs

Cf. A002193 (decimal expansion), A006784 (for definition of Engel expansion), A028257 (Engel expansion of sqrt(3)).

Programs

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 *)

Extensions

More terms from Simon Plouffe, Jan 05 2002