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A028259
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Engel expansion of the golden ratio, (1 + sqrt(5))/2 = 1.61803... .
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5
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1, 2, 5, 6, 13, 16, 16, 38, 48, 58, 104, 177, 263, 332, 389, 4102, 4575, 5081, 9962, 18316, 86613, 233239, 342534, 964372, 1452850, 7037119, 7339713, 8270361, 12855437, 15900982, 19211148, 1365302354, 1565752087, 1731612283
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OFFSET
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1,2
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COMMENTS
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Cf. A006784 for definition of Engel expansion.
(sqrt(5) - 1)/2 = 1/(golden ratio) has the same Engel expansion starting with a(2). - G. C. Greubel, Oct 16 2016
Each Engel expansion for a constant x (x > 0) starts with floor(x) 1's. After that, the mean growth of the terms is by a factor e for most constants x, i.e., the order of magnitude of the n-th entry is exp(n-floor(c)), for n large enough. This comment is similar to the fact that for the continued fraction terms of most constants, the geometric mean of those terms equals the Khintchine constant for n large enough. Moreover, note that for the golden section, all continued fraction terms are 1 and thus do not obey the Gauss-Kuzmin distribution which leads to the Khintchine constant (i.e., the Khintchine measure for the golden ratio is 1), but the Engel expansion does obey the statistic behavior for most constants. - A.H.M. Smeets, Aug 24 2018
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REFERENCES
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F. Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191.
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LINKS
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A.H.M. Smeets, Table of n, a(n) for n = 1..2406 (Terms 1 through 300 from T. D. Noe, 301 through 698 from Simon Plouffe, and 699 through 1500 from G. C. Greubel)
F. Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191. English translation by Georg Fischer, included with his permission.
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FORMULA
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Limit_{n -> oo} log(a(n + floor(golden ratio)))/n = 1.
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MATHEMATICA
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EngelExp[ 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 ] ]
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PROG
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(Python)
j = 0
while j<3100000:
# to obtain n correct Engel expansion terms about n^2/2 continued fraction steps are needed; 3100000 is a safe bound
....if j == 0:
........p0, q0 = 1, 1
....elif j == 1:
........p1, q1 = p0+1, 1
....else:
........p0, p1 = p1, p1+p0
........q0, q1 = q1, q1+q0
....j = j+1
i = 0
while i < 2410:
....i = i+1
....a = q0//p0+1
....print(i, a)
....p0 = a*p0-q0
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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Naoki Sato (naoki(AT)math.toronto.edu), Dec 11 1999
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EXTENSIONS
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STATUS
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approved
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