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A016627 Decimal expansion of log(4). 16
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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-Fran├žois 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

REFERENCES

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.

LINKS

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.

FORMULA

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

EXAMPLE

1.38629436111989061883446424291635313615100026872051050824136...

MATHEMATICA

RealDigits[Log@ 4, 10, 111][[1]] (* Robert G. Wilson v, Aug 31 2014 *)

PROG

(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

CROSSREFS

Cf. A016732 (continued fraction).

Sequence in context: A021263 A246727 A081803 * A175184 A019604 A214726

Adjacent sequences:  A016624 A016625 A016626 * A016628 A016629 A016630

KEYWORD

nonn,cons

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified December 9 06:30 EST 2016. Contains 278963 sequences.