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A028254
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Engel expansion of sqrt(2).
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7
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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
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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...
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MATHEMATICA
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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 *)
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CROSSREFS
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Cf. A002193 (decimal expansion), A006784 (for definition of Engel expansion), A028257 (Engel expansion of sqrt(3)).
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KEYWORD
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nonn
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AUTHOR
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Naoki Sato (naoki(AT)math.toronto.edu)
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EXTENSIONS
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STATUS
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approved
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