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A002390 Decimal expansion of natural logarithm of golden ratio.
(Formerly M3318 N1334)
29
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

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

REFERENCES

Mohammad K. Azarian, Problem 123, Missouri Journal of Mathematical Sciences, Vol. 10, No. 3, Fall 1998, p. 176. Solution published in Vol. 12, No. 1, Winter 2000, pp. 61-62.

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.

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.

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

FORMULA

Also equals asinh(1/2).

Equals sqrt(5)/2 * Sum_{n>=1} (-1)^(n-1)/(n*C(2*n,n)). - Seiichi Kirikami, Aug 20 2011

Equals sqrt(5)/4*(5*(Sum_{n>=1} (-1)^(n-1)/C(2*n,n))-1). - 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)). - 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)), 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

EXAMPLE

0.481211825059603447497758913424368423135184334385660519661...

MAPLE

arcsinh(1/2);

MATHEMATICA

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

PROG

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

CROSSREFS

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

Sequence in context: A127734 A154520 A244690 * A193087 A201404 A122149

Adjacent sequences:  A002387 A002388 A002389 * A002391 A002392 A002393

KEYWORD

nonn,cons

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified March 30 18:30 EDT 2017. Contains 284302 sequences.