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A028254 Engel expansion of sqrt(2). 7
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 (list; graph; refs; listen; history; text; internal format)
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

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

Sequence in context: A188345 A028265 A084041 * A318351 A137780 A079372

Adjacent sequences:  A028251 A028252 A028253 * A028255 A028256 A028257

KEYWORD

nonn

AUTHOR

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

EXTENSIONS

More terms from Simon Plouffe, Jan 05 2002

STATUS

approved

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Last modified July 13 22:21 EDT 2020. Contains 335716 sequences. (Running on oeis4.)