A028259 Engel expansion of the golden ratio, (1 + sqrt(5))/2 = 1.61803... .

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

Offset

1,2

Comments

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

References

F. Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191.

Links

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.

P. Erdős and Jeffrey Shallit, New bounds on the length of finite Pierce and Engel series, Sém. Théor. Nombres Bordeaux (2) 3 (1991), no. 1, 43-53.

Eric Weisstein's World of Mathematics, Engel Expansion

Eric Weisstein's World of Mathematics, Golden Ratio

Index entries for sequences related to Engel expansions

Formula

Limit_{n -> oo} log(a(n + floor(golden ratio)))/n = 1.

Mathematica

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 ] ]

Prog

(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
# A.H.M. Smeets, Aug 24 2018

Crossrefs

Cf. A001622, A220398.

Sequence in context: A113240 A098376 A342569 * A327472 A283684 A325285

Adjacent sequences: A028256 A028257 A028258 * A028260 A028261 A028262

Keyword

nonn,easy,nice

Author

Naoki Sato (naoki(AT)math.toronto.edu), Dec 11 1999

Extensions

Corrected and extended by Vladeta Jovovic, Aug 16 2001

Status

approved