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A002391
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Decimal expansion of natural logarithm of 3.
(Formerly M4595 N1960)
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16
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1, 0, 9, 8, 6, 1, 2, 2, 8, 8, 6, 6, 8, 1, 0, 9, 6, 9, 1, 3, 9, 5, 2, 4, 5, 2, 3, 6, 9, 2, 2, 5, 2, 5, 7, 0, 4, 6, 4, 7, 4, 9, 0, 5, 5, 7, 8, 2, 2, 7, 4, 9, 4, 5, 1, 7, 3, 4, 6, 9, 4, 3, 3, 3, 6, 3, 7, 4, 9, 4, 2, 9, 3, 2, 1, 8, 6, 0, 8, 9, 6, 6, 8, 7, 3, 6, 1, 5, 7, 5, 4, 8, 1, 3, 7, 3, 2, 0, 8, 8, 7, 8, 7, 9, 7
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OFFSET
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1,3
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REFERENCES
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W. E. Mansell, Tables of Natural and Common Logarithms. Royal Society Mathematical Tables, Vol. 8, Cambridge Univ. Press, 1964, p. 2.
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.
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).
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LINKS
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Harry J. Smith, Table of n, a(n) for n = 1..20000
Eric Weisstein's World of Mathematics, BBP-Type Formula
D. H. Bailey, Compendium to BBP formulas
G. Huvent, Formules BBP en base 3 [From Jaume Oliver Lafont, Oct 12 2009]
_Simon Plouffe_, Plouffe's Inverter, The natural logarithm of 3 to 10000 digits
_Simon Plouffe_, log(3), natural logarithm of 3 to 2000 places
S. Ramanujan, Notebook entry
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FORMULA
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log(3) = sum_{n>=1} (9*n-4)/((3*n-2)*(3*n-1)*3*n). [Jolley, Summation of Series, Dover (1961) eq 74]
ln(3) = 1/4*(1+ Sum((1/(9)^(k+1))*(27/(2*k+1) + 4/(2*k+2) + 1/(2*k+3)), k = 0 .. infinity) ) (a BBP-type formula) [From Alexander R. Povolotsky, Dec 01 2008]
log(3) = 4/5 +2/10*sum((1/4)^n*(1/(2*n+1)+1/(2*n+3)),n=0...infinity) [From Alexander R. Povolotsky, Dec 18 2008]
log(3)=sum((1/9)^(k+1)(9/(2k+1)+1/(2k+2)),k=0..infinity) [From Jaume Oliver Lafont, Dec 22 2008]
Sum_{i>=1} 1/(9^i*i) + Sum_{i>=0} 1/(9^i*(i+1/2)) = 2*log(3) (Huvent 2001) [From Jaume Oliver Lafont, Oct 12 2009]
log(3) = sum(k>=1, A191907(3,k)/k ). (conjecture) [From Mats Granvik, Jun 19 2011]
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EXAMPLE
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1.098612288668109691395245236922525704647490557822749451734694333637494...
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MATHEMATICA
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RealDigits[Log[3], 10, 120][[1]] (* From Harvey P. Dale, Apr 23 2011 *)
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PROG
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(PARI) log(3) \\ Charles R Greathouse IV, Jan 24 2012
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CROSSREFS
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Cf. A058962, A154920, A002162, A016731 (continued fraction).
Sequence in context: A059068 A059069 A084660 * A193626 A087044 A105415
Adjacent sequences: A002388 A002389 A002390 * A002392 A002393 A002394
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KEYWORD
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nonn,cons,changed
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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Editing and more terms from Charles R Greathouse IV, Apr 20 2010
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STATUS
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approved
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