%I A002162 M4074 N1689
%S A002162 6,9,3,1,4,7,1,8,0,5,5,9,9,4,5,3,0,9,4,1,7,2,3,2,1,2,1,4,5,8,1,7,6,
%T A002162 5,6,8,0,7,5,5,0,0,1,3,4,3,6,0,2,5,5,2,5,4,1,2,0,6,8,0,0,0,9,4,9,3,
%U A002162 3,9,3,6,2,1,9,6,9,6,9,4,7,1,5,6,0,5,8,6,3,3,2,6,9,9,6,4,1,8,6,8,7
%N A002162 Decimal expansion of natural logarithm of 2.
%C A002162 ln(2) = 1/4*(3 - sum(1/(n*(n+1)*(2*n+1)),n=1...infinity)) - from and 
               by Alexander R. Povolotsky [From Alexander R. Povolotsky (pevnev(AT)juno.com), 
               Dec 16 2008]
%C A002162 ln(2) = 105*(sum(1/(2*n*(2*n+1)*(2*n+3)*(2*n+5)*(2*n+7)),n=1...infinity) 
               - 319/44100) ln(2) = (319/420 - 3/2*sum(1/(6*n^2+39*n+63),n=1...infinity)) 
               [From Alexander R. Povolotsky (pevnev(AT)juno.com), Dec 16 2008]
%C A002162 i^2*ln(1/2) [From Eric Desbiaux (moongerms(AT)wanadoo.fr), Jun 27 2009]
%D A002162 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A002162 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002162 S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.3.3.
%D A002162 D. W. Sweeney, On the computation of Euler's constant, Math. Comp., 17 
               (1963), 170-178.
%D A002162 Uhler, Horace S.; Recalculation and extension of the modulus and of the 
               logarithms of 2, 3, 5, 7 and 17. Proc. Nat. Acad. Sci. U. S. A. 26, 
               (1940). 205-212.
%H A002162 Harry J. Smith, <a href="b002162.txt">Table of n, a(n) for n=0,...,20000</
               a>
%H A002162 D. H. Bailey and J. M. Borwein, <a href="http://eprints.cecm.sfu.ca/archive/
               00000269/">Experimental Mathematics: Examples, Methods and Implications</
               a>
%H A002162 Paul Cooijmans, <a href="http://web.archive.org/web/20050302174449/http:/
               /members.chello.nl/p.cooijmans/gliaweb/tests/odds.html">Odds</a>.
%H A002162 X. Gourdon and P. Sebah, <a href="http://numbers.computation.free.fr/
               Constants/Log2/log2.html">The logarithm constant:log(2)</a>
%H A002162 S. Plouffe, <a href="http://www.worldwideschool.org/library/books/sci/
               math/MiscellaneousMathematicalConstants/chap58.html">log(2), natural 
               logarithm of 2 to 2000 places</a>
%H A002162 S. Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/
               question/q260.htm">Question 260</a>, J. Ind. Math. Soc.
%H A002162 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               NaturalLogarithmof2.html">Natural Logarithm of 2</a>
%H A002162 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Masser-GramainConstant.html">Masser-Gramain Constant</a>
%H A002162 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               LogarithmicIntegral.html">Logarithmic Integral</a>
%F A002162 log(2) = Sum_{ k >= 1 } 1/(k*2^k) = Sum_{j >= 1} (-1)^(j+1)/j.
%F A002162 log(2) = Integral_{t = 0..1 } dt/(1+t).
%F A002162 log(2) = 2/3 * (1 + Sum{k=1..inf, 2/[(4k)^3-4k]}) (Ramanujan).
%F A002162 log(2)=4*sum_{k=0..inf} [3-2*sqrt(2)]^(2k+1)/(2k+1) (Y. Luke) - R. J. 
               Mathar (mathar(AT)strw.leidenuniv.nl), Jul 13 2006
%F A002162 log(2) = 1-(1/2)Sum_{ k >= 1 } 1/(k*(2k+1)) [From Jaume Oliver Lafont 
               (joliverlafont(AT)gmail.com), Jan 06 2009, Jan 08 2009]
%F A002162 log(2) = 4*Sum_{ k >= 0 } 1/((4k+1)(4k+2)(4k+3)) [From Jaume Oliver Lafont 
               (joliverlafont(AT)gmail.com), Jan 08 2009]
%F A002162 Equals 7/12+24*sum_{k=1..infinity} 1/(A052787(k+4)*A000079(k)). [From 
               R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 23 2009]
%F A002162 Beginning of the contribution from Alexander R. Povolotsky In addition 
               to formula posted by me in Program/Maple section on Dec 16, 2008 
               ln(2) = 1/4*(3 - sum(1/(n*(n+1)*(2*n+1)), n=1...infinity)) Here is 
               the BBP style formula I came up with today ln(2)=(230166911/9240-Sum((1/
               2)^k* (11/k+10/(k+1)+9/(k+2)+8/(k+3)+7/(k+4)+6/(k+5)-6/(k+7)-7/(k+8)-8/
               (k+9) -9/(k+10)-10/(k+11)), k = 1 .. infinity))/35917 End of the 
               contribution from Alexander R. Povolotsky [From Alexander R. Povolotsky 
               (pevnev(AT)juno.com), Jul 04 2009]
%F A002162 log(2) = A052882/A000670. [From Mats Granvik (mats.granvik(AT)abo.fi), 
               Aug 10 2009]
%F A002162 From log(1-x-x^2) at x=1/2, log(2)=(1/2)*Sum_{k>=1}L(k)/(k*2^k), where 
               L(n) is the n-th Lucas number (A000032). [From Jaume Oliver Lafont 
               (joliverlafont(AT)gmail.com), Oct 24 2009]
%e A002162 .6931471805599453...
%e A002162 0.693147180559945309417232121458176568075500134360255254120680009493393... 
               [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), May 16 2009]
%p A002162 ln(2) = 1/4*(3 - sum(1/(n*(n+1)*(2*n+1)),n=1...infinity)) [From Alexander 
               R. Povolotsky (pevnev(AT)juno.com), Dec 16 2008]
%o A002162 (PARI) { default(realprecision, 20080); x=10*log(2); for (n=0, 20000, 
               d=floor(x); x=(x-d)*10; write("b002162.txt", n, " ", d)); } [From 
               Harry J. Smith (hjsmithh(AT)sbcglobal.net), Apr 21 2009]
%Y A002162 Cf. A016730 Continued fraction. [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), 
               May 16 2009]
%Y A002162 Sequence in context: A129938 A022698 A013707 this_sequence A072365 A085138 
               A153872
%Y A002162 Adjacent sequences: A002159 A002160 A002161 this_sequence A002163 A002164 
               A002165
%K A002162 cons,nonn
%O A002162 0,1
%A A002162 N. J. A. Sloane (njas(AT)research.att.com).
%E A002162 Fixed my PARI program, had -n Harry J. Smith (hjsmithh(AT)sbcglobal.net), 
               May 19 2009