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%I #115 Mar 21 2024 07:03:31
%S 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,
%T 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,
%U 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
%N Decimal expansion of log(4).
%C This constant (negated) is the 1-dimensional analog of Madelung's constant. - _Jean-François Alcover_, May 20 2014
%C 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
%C log(4) - 1 is the mean ratio between the smaller length and the larger length of the two parts of a stick that is being broken at a point that is uniformly chosen at random (Mosteller, 1965). - _Amiram Eldar_, Jul 25 2020
%C From _Bernard Schott_, Sep 11 2020: (Start)
%C This constant was the subject of the problem B5 during the 42nd Putnam competition in 1981 (see formula Sep 11 2020 and Putnam link).
%C Jeffrey Shallit generalizes this result obtained for base 2 to any base b (see Amer. Math. Month. link): Sum_{k>=1} digsum(k)_b / (k*(k+1)) = (b/(b-1)) * log(b) where digsum(k)_b is the sum of the digits of k when expressed in base b (for base 10 see A334388). (End)
%D 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.
%D Frederick Mosteller, Fifty challenging problems of probability, Dover, New York, 1965. See problem 42, pp. 10 and 63.
%H Harry J. Smith, <a href="/A016627/b016627.txt">Table of n, a(n) for n = 1..20000</a>
%H Milton Abramowitz and Irene A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H Lawrence Downey, Boon W. Ong and James A. Sellers, <a href="https://www.d.umn.edu/~jsellers/downey_ong_sellers_cmj_preprint.pdf">Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers</a>, Coll. Math. J., 39, no. 5 (2008), 391-394.
%H D. H. Lehmer, <a href="http://www.jstor.org/stable/2322496">Interesting series involving the Central Binomial Coefficient</a>, Am. Math. Monthly, Vol. 92, No. 7 (1985) pp. 449-457.
%H Allon G. Percus, Gabriel Istrate, Bruno Goncalves, Robert Z. Sumi and Stefan Boettcher, <a href="http://arxiv.org/abs/0808.1549">The Peculiar Phase Structure of Random Graph Bisection</a>, arXiv:0808.1549 [cond-mat.stat-mech], 2008.
%H H.-J. Seiffert, <a href="https://fq.math.ca/Scanned/32-4/elementary32-4.pdf">Problem B-771</a>, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 32, No. 4 (1994), p. 374; <a href="https://www.fq.math.ca/Scanned/33-5/elementary33-5.pdf">More Sums</a>, Solution to Problem B-771 by Don Redmond, ibid., Vol. 33, No. 5 (1995), pp. 470-471.
%H J. O. Shallit, <a href="https://www.jstor.org/stable/2322523">Solutions of Advanced Problems, 6450</a>, The American Mathematical Monthly, Vol. 92, No. 7, Aug.-Sep., 1985, pp. 513-514; DOI: 10.2307/2322523.
%H 42nd Putnam Competition, <a href="https://prase.cz/kalva/putnam/putn81.html">Problem B5</a>, 1981.
%H <a href="/index/O#Olympiads">Index entries for sequences related to Olympiads and other Mathematical competitions</a>.
%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%F log(4) = Sum_{k >= 1} H(k)/2^k where H(k) is the k-th harmonic number. - _Benoit Cloitre_, Jun 15 2003
%F 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
%F Equals 2*A002162 = Sum_{n >= 1} binomial(2*n, n)/(n*4^n) [D. H. Lehmer, Am. Math. Monthly 92 (1985) 449 and Jolley eq. 262]. - _R. J. Mathar_, Mar 04 2009
%F log(4) = Sum_{k >= 1} A191907(4, k)/k, (conjecture). - _Mats Granvik_, Jun 19 2011
%F log(4) = lim_{n -> infinity} A066066(n)/n. - _M. F. Hasler_, Oct 20 2013
%F Equals Sum_{k >= 1} 1/( 2*k^2 - k ). - _Robert G. Wilson v_, Aug 31 2014
%F 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
%F From _Amiram Eldar_, Jul 25 2020: (Start)
%F Equals Sum_{k>=1} (3/4)^k/k.
%F Equals Sum_{k>=1} 1/(k*2^(k-1)) = Sum_{k>=1} 1/A001787(k).
%F Equals Integral_{x=0..1} log(1+1/x) dx. (End)
%F Equals Sum_{k>=1} A000120(k) / (k*(k+1)). - _Bernard Schott_, Sep 11 2020
%F Equals 1 + Sum_{k>=1} zeta(2*k+1)/4^k. - _Amiram Eldar_, May 27 2021
%F Equals Sum_{k>=1} (2*k+1)*Fibonacci(k)/(k*(k+1)*2^k) (Seiffert, 1994). - _Amiram Eldar_, Jan 15 2022
%F Continued fraction: log(4) = 1 + 1/(2 + (1*2)/(2 + (2*3)/(2 + (3*4)/(2 + (4*5)/(2 + ... ))))) due to Euler. - _Peter Bala_, Mar 05 2024
%F log(4) = 2*Sum_{n >= 1} 1/(n*P(n, 5/3)*P(n-1, 5/3)), where P(n, x) denotes the n-th Legendre polynomial. The first 20 terms of the series gives the approximation log(4) = 1.386294361119890618(66...), correct to 18 decimal places. - _Peter Bala_, Mar 18 2024
%e 1.38629436111989061883446424291635313615100026872051050824136...
%t RealDigits[Log@ 4, 10, 111][[1]] (* _Robert G. Wilson v_, Aug 31 2014 *)
%o (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
%o (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
%Y Cf. A016732 (continued fraction).
%Y Cf. A002162 (half), A133362 (reciprocal).
%Y Cf. A000384, A001787, A002378, A007531, A066066, A069072, A191907.
%Y Cf. A000045, A000120, A334388.
%K nonn,cons
%O 1,2
%A _N. J. A. Sloane_
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