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A002390 Decimal expansion of natural logarithm of golden ratio.

%I A002390 M3318 N1334 #148 Feb 16 2025 08:32:25
%S A002390 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,
%T A002390 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,
%U A002390 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
%N A002390 Decimal expansion of natural logarithm of golden ratio.
%C A002390 The Baxa article proves that every gamma >= this constant is the Lévy constant of a transcendental number. - _Michel Marcus_, Apr 09 2016
%C A002390 The entropy of the golden mean shift. See Capobianco link. - _Michel Marcus_, Jan 19 2019
%C A002390 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
%D A002390 George Boros and Victor H. Moll, Irresistible integrals, Cambridge University Press (2006), p. 236.
%D A002390 W. E. Mansell, Tables of Natural and Common Logarithms. Royal Society Mathematical Tables, Vol. 8, Cambridge Univ. Press, 1964, p. XVIII.
%D A002390 B. Muslu, Sayılar ve Bağlantılar 2, Luna, 2021, pages 31-38.
%D A002390 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002390 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002390 G. C. Greubel, <a href="/A002390/b002390.txt">Table of n, a(n) for n = 0..5000</a>
%H A002390 Alexander Adamchuk's comment, Sep 01 2006 <a href="http://ru-math.livejournal.com/399814.html">Mathematics in Russian</a>
%H A002390 Christoph Baxa, <a href="http://dx.doi.org/10.1090/S0002-9939-09-09787-1">Lévy constants of transcendental numbers</a>, Proc. Amer. Math. Soc. 137 (2009), 2243-2249.
%H A002390 Christopher Brown, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/55-5/Brown.pdf">The natural logarithm of the golden section</a>, Fibonacci Quarterly 55:5 (2017), pp. 42-44.
%H A002390 Silvio Capobianco, <a href="http://cs.ioc.ee/~silvio/slides/sd4.pdf">Introduction to Symbolic Dynamics. Part 4: Entropy; The entropy of the golden mean shift</a>, Institute of Cybernetics at TUT; May 12 2010. Slides 15-17.
%H A002390 Simon Plouffe, Plouffe's Inverter, <a href="http://www.plouffe.fr/simon/constants/logphi.txt">ln(phi) to 10000 digits</a>
%H A002390 Simon Plouffe, <a href="https://web.archive.org/web/20080205212743/http://www.worldwideschool.org/library/books/sci/math/MiscellaneousMathematicalConstants/chap65.html">ln(0.5+0.5*SQRT(5)) to 2000 digits</a>
%H A002390 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FibonacciHyperbolicFunctions.html">Fibonacci Hyperbolic Functions</a>
%H A002390 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F A002390 Also equals arcsinh(1/2).
%F A002390 Equals sqrt(5)* A086466 /2. - _Seiichi Kirikami_, Aug 20 2011
%F A002390 Equals sqrt(5)*(5* A086465 -1)/4. - _Jean-François Alcover_, Apr 29 2013
%F A002390 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
%F A002390 Equals sqrt(5)/4 * Sum_{n>=0} (-1)^n/((2n+1)*C(2*n,n)) = sqrt(5) *A344041 /4. - _Alexander Adamchuk_, Dec 27 2013
%F A002390 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
%F A002390 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
%F A002390 Equals Sum_{k>=1} cos(Pi*k/5)/k. - _Amiram Eldar_, Aug 12 2020
%F A002390 Equals real solution to exp(x)+exp(2*x) = exp(3*x). - _Alois P. Heinz_, Jul 14 2022
%F A002390 Equals arccoth(sqrt(5)). - _Amiram Eldar_, Feb 09 2024
%F A002390 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
%F A002390 Equals Sum_{n>=0} ((-1)^(n)*binomial(2*n, n))/(2^(4*n + 1)*(2*n + 1)). - _Antonio Graciá Llorente_, Nov 13 2024
%e A002390 0.481211825059603447497758913424368423135184334385660519661...
%p A002390 arcsinh(1/2); evalf(%, 120);
%t A002390 RealDigits[N[Log[GoldenRatio],200]][[1]] (* _Vladimir Joseph Stephan Orlovsky_, Feb 20 2011 *)
%o A002390 (PARI) asinh(1/2) \\ _Charles R Greathouse IV_, Jan 04 2016
%Y A002390 Cf. A000108, A001622, A013661, A086463, A086466, A263401.
%K A002390 nonn,cons
%O A002390 0,1
%A A002390 _N. J. A. Sloane_