A002390 Decimal expansion of natural logarithm of golden ratio.

4, 8, 1, 2, 1, 1, 8, 2, 5, 0, 5, 9, 6, 0, 3, 4, 4, 7, 4, 9, 7, 7, 5, 8, 9, 1, 3, 4, 2, 4, 3, 6, 8, 4, 2, 3, 1, 3, 5, 1, 8, 4, 3, 3, 4, 3, 8, 5, 6, 6, 0, 5, 1, 9, 6, 6, 1, 0, 1, 8, 1, 6, 8, 8, 4, 0, 1, 6, 3, 8, 6, 7, 6, 0, 8, 2, 2, 1, 7, 7, 4, 4, 1, 2, 0, 0, 9, 4, 2, 9, 1, 2, 2, 7, 2, 3, 4, 7, 4

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• cons

Comments

The Baxa article proves that every gamma >= this constant is the Lévy constant of a transcendental number. - Michel Marcus, Apr 09 2016

The entropy of the golden mean shift. See Capobianco link. - Michel Marcus, Jan 19 2019

Also the limiting value of the area of the function y = 1/x bounded by the abscissa of consecutive F(n) points (where F(n)=A000045(n) are the Fibonacci numbers and n > 0). - Burak Muslu, May 09 2021

Examples

			0.481211825059603447497758913424368423135184334385660519661...
		

References

• George Boros and Victor H. Moll, Irresistible integrals, Cambridge University Press (2006), p. 236.
• W. E. Mansell, Tables of Natural and Common Logarithms. Royal Society Mathematical Tables, Vol. 8, Cambridge Univ. Press, 1964, p. XVIII.
• B. Muslu, Sayılar ve Bağlantılar 2, Luna, 2021, pages 31-38.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• Alexander Adamchuk's comment, Sep 01 2006 Mathematics in Russian
• Christoph Baxa, Lévy constants of transcendental numbers, Proc. Amer. Math. Soc. 137 (2009), 2243-2249.
• Christopher Brown, The natural logarithm of the golden section, Fibonacci Quarterly 55:5 (2017), pp. 42-44.
• Silvio Capobianco, Introduction to Symbolic Dynamics. Part 4: Entropy; The entropy of the golden mean shift, Institute of Cybernetics at TUT; May 12 2010. Slides 15-17.
• Simon Plouffe, Plouffe's Inverter, ln(phi) to 10000 digits
• Simon Plouffe, ln(0.5+0.5*SQRT(5)) to 2000 digits
• Eric Weisstein's World of Mathematics, Fibonacci Hyperbolic Functions
• Index entries for transcendental numbers

Crossrefs

Cf. A000108, A001622, A013661, A086463, A086466, A263401.

Programs

Maple

arcsinh(1/2); evalf(%, 120);

Mathematica

RealDigits[N[Log[GoldenRatio],200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2011 *)

PARI

asinh(1/2) \\ Charles R Greathouse IV, Jan 04 2016

Formula

Also equals arcsinh(1/2).

Equals sqrt(5)* A086466 /2. - Seiichi Kirikami, Aug 20 2011

Equals sqrt(5)*(5* A086465 -1)/4. - Jean-François Alcover, Apr 29 2013

Also equals (125*C - 55) / (24*sqrt(5)), where C = Sum_{k>=1} (-1)^(k+1)*1/Cat(k), where Cat(k) = (2k)!/k!/(k+1)! = A000108(k) - k-th Catalan number. See Sep 01 2006 comment at ref. Mathematics in Russian. - Alexander Adamchuk, Dec 27 2013

Equals sqrt(5)/4 * Sum_{n>=0} (-1)^n/((2n+1)*C(2*n,n)) = sqrt(5) *A344041 /4. - Alexander Adamchuk, Dec 27 2013

Equals sqrt((Pi^2/6 - W)/3), where W = Sum_{n>=0} (-1)^n/((2n+1)^2*C(2*n,n)) = A145436, attributed by Alexander Adamchuk to Ramanujan. See Sep 01 2006 comment at ref. Mathematics in Russian. - Alexander Adamchuk, Dec 27 2013

Equals lim_{j->infinity} Sum_{k=F(j)..F(j+1)-1} (1/k), where F = A000045, the Fibonacci sequence. Convergence is slow. For example: Sum_{k=21..33} (1/k) = 0.4910585.... - Richard R. Forberg, Aug 15 2014

Equals Sum_{k>=1} cos(Pi*k/5)/k. - Amiram Eldar, Aug 12 2020

Equals real solution to exp(x)+exp(2*x) = exp(3*x). - Alois P. Heinz, Jul 14 2022

Equals arccoth(sqrt(5)). - Amiram Eldar, Feb 09 2024

Sum_{n >= 1} 1/(n*P(n, sqrt(5))*P(n-1, sqrt(5))), where P(n, x) denotes the n-th Legendre polynomial. The first ten terms of the series gives the approximation log((1 + sqrt(5))/2) = 0.481211825059(39..), correct to 12 decimal places. - Peter Bala, Mar 16 2024

Equals Sum_{n>=0} ((-1)^(n)*binomial(2*n, n))/(2^(4*n + 1)*(2*n + 1)). - Antonio Graciá Llorente, Nov 13 2024