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

Author

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

Keywords

• nonn
• easy
• nice

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

Crossrefs

Cf. A001622, A220398.

Programs

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 ] ]

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

Formula

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

Extensions

Corrected and extended by Vladeta Jovovic, Aug 16 2001