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A016627
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Decimal expansion of log(4).
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20
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1, 3, 8, 6, 2, 9, 4, 3, 6, 1, 1, 1, 9, 8, 9, 0, 6, 1, 8, 8, 3, 4, 4, 6, 4, 2, 4, 2, 9, 1, 6, 3, 5, 3, 1, 3, 6, 1, 5, 1, 0, 0, 0, 2, 6, 8, 7, 2, 0, 5, 1, 0, 5, 0, 8, 2, 4, 1, 3, 6, 0, 0, 1, 8, 9, 8, 6, 7, 8, 7, 2, 4, 3, 9, 3, 9, 3, 8, 9, 4, 3, 1, 2, 1, 1, 7, 2, 6, 6, 5, 3, 9, 9, 2, 8, 3, 7, 3, 7
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OFFSET
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1,2
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COMMENTS
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Constant cited in the Percus reference. - Jonathan Vos Post, Aug 13 2008
This constant (negated) is also the 1-dimensional analog of Madelung's constant. - Jean-François Alcover, May 20 2014
This constant is the sum over the reciprocals of the hexagonal numbers A000384(n), n >= 1. See the Downey et al. link, and the formula by Robert G. Wilson v below. - Wolfdieter Lang, Sep 12 2016
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REFERENCES
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Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 2.
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LINKS
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Harry J. Smith, Table of n, a(n) for n = 1..20000
Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Lawrence Downey, Boon W. Ong, and James A. Sellers, Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers, Coll. Math. J., 39, no. 5 (2008), 391-394.
Allon G. Percus, Gabriel Istrate, Bruno Goncalves, Robert Z. Sumi and Stefan Boettcher, The Peculiar Phase Structure of Random Graph Bisection, arXiv:0808.1549 [cond-mat.stat-mech], 2008.
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FORMULA
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log(4) = sum(k >= 1, H(k)/2^k) where H(k) is the k-th harmonic number. - Benoit Cloitre, Jun 15 2003
Equals 1-sum_{k >= 1} (-1)^k/A002378(k) = 1 + 2*sum_{k >= 0} 1/A069072(k) = 5/4 - sum_{k >= 1} (-1)^k/A007531(k+2). - R. J. Mathar, Jan 23 2009
Equals 2*A002162 = sum_{n >= 1} binomial(2n, n)/(n*4^n) [D. H. Lehmer, Am. Math. Monthly 92 (1985) 449 and Jolley eq. 262]. - R. J. Mathar, Mar 04 2009
log(4) = sum(k >= 1, A191907(4, k)/k ), (conjecture). - Mats Granvik, Jun 19 2011
log(4) = lim_{n -> infinity} A066066(n)/n. - M. F. Hasler, Oct 20 2013
Equals sum_{k >= 1} 1/( 2*k^2 - k ). - Robert G. Wilson v, Aug 31 2014
Equals gamma(0, 1/2) - gamma(0, 1) = -(EulerGamma + polygamma(0, 1/2)), where gamma(n,x) denotes the generalized Stieltjes constants. - Peter Luschny, May 16 2018
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EXAMPLE
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1.38629436111989061883446424291635313615100026872051050824136...
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MATHEMATICA
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RealDigits[Log@ 4, 10, 111][[1]] (* Robert G. Wilson v, Aug 31 2014 *)
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PROG
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(PARI) { default(realprecision, 20080); x=log(4); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b016627.txt", n, " ", d)); } \\ Harry J. Smith, May 16 2009, corrected May 19 2009
(PARI) A016627_vec(N)=digits(floor(log(precision(4., N))*10^(N-1))) \\ Or: default(realprecision, N); digits(log(4)\.1^N) \\ M. F. Hasler, Oct 20 2013
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CROSSREFS
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Cf. A016732 (continued fraction).
Sequence in context: A021263 A246727 A081803 * A175184 A019604 A214726
Adjacent sequences: A016624 A016625 A016626 * A016628 A016629 A016630
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KEYWORD
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nonn,cons
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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