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

%I A028259 #87 Feb 16 2025 08:32:35
%S A028259 1,2,5,6,13,16,16,38,48,58,104,177,263,332,389,4102,4575,5081,9962,
%T A028259 18316,86613,233239,342534,964372,1452850,7037119,7339713,8270361,
%U A028259 12855437,15900982,19211148,1365302354,1565752087,1731612283
%N A028259 Engel expansion of the golden ratio, (1 + sqrt(5))/2 = 1.61803... .
%C A028259 Cf. A006784 for definition of Engel expansion.
%C A028259 (sqrt(5) - 1)/2 = 1/(golden ratio) has the same Engel expansion starting with a(2). - _G. C. Greubel_, Oct 16 2016
%C A028259 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
%D A028259 F. Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191.
%H A028259 A.H.M. Smeets, <a href="/A028259/b028259.txt">Table of n, a(n) for n = 1..2406</a> (Terms 1 through 300 from T. D. Noe, 301 through 698 from Simon Plouffe, and 699 through 1500 from G. C. Greubel)
%H A028259 F. Engel, <a href="/A006784/a006784.pdf">Entwicklung der Zahlen nach Stammbruechen</a>, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191. English translation by Georg Fischer, included with his permission.
%H A028259 P. Erdős and Jeffrey Shallit, <a href="http://www.numdam.org/item?id=JTNB_1991__3_1_43_0">New bounds on the length of finite Pierce and Engel series</a>, Sém. Théor. Nombres Bordeaux (2) 3 (1991), no. 1, 43-53.
%H A028259 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EngelExpansion.html">Engel Expansion</a>
%H A028259 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GoldenRatio.html">Golden Ratio</a>
%H A028259 <a href="/index/El#Engel">Index entries for sequences related to Engel expansions</a>
%F A028259 Limit_{n -> oo} log(a(n + floor(golden ratio)))/n = 1.
%t A028259 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 ] ]
%o A028259 (Python)
%o A028259 j = 0
%o A028259 while j<3100000:
%o A028259 # to obtain n correct Engel expansion terms about n^2/2 continued fraction steps are needed; 3100000 is a safe bound
%o A028259     if j == 0:
%o A028259         p0, q0 = 1, 1
%o A028259     elif j == 1:
%o A028259         p1, q1 = p0+1, 1
%o A028259     else:
%o A028259         p0, p1 = p1, p1+p0
%o A028259         q0, q1 = q1, q1+q0
%o A028259     j = j+1
%o A028259 i = 0
%o A028259 while i < 2410:
%o A028259     i = i+1
%o A028259     a = q0//p0+1
%o A028259     print(i, a)
%o A028259     p0 = a*p0-q0
%o A028259 # _A.H.M. Smeets_, Aug 24 2018
%Y A028259 Cf. A001622, A220398.
%K A028259 nonn,easy,nice
%O A028259 1,2
%A A028259 Naoki Sato (naoki(AT)math.toronto.edu), Dec 11 1999
%E A028259 Corrected and extended by _Vladeta Jovovic_, Aug 16 2001