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Search: id:A002391
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| A002391 |
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Decimal expansion of natural logarithm of 3.
(Formerly M4595 N1960)
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+0
10
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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
(list; cons; graph; listen)
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OFFSET
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1,3
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COMMENT
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Two BBP-type Formulas are provided in http://www.mathworld.wolfram.com/BBP-TypeFormula.html Also here is Alexander Povolotsky's alternative formula for ln(3), which is not listed in http://www.mathworld.wolfram.com/BBP-TypeFormula.html 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) ) [From Alexander R. Povolotsky (pevnev(AT)juno.com), Dec 01 2008]
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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
D. H. Bailey, Compendium to BBP formulas
G. Huvent, Formules BBP en base 3 [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Oct 12 2009]
S. Plouffe, Plouffe's Inverter, The natural logarithm of 3 to 10000 digits
S. Plouffe, log(3), natural logarithm of 3 to 2000 places
S. Ramanujan, Notebook entry
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FORMULA
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Another formula from and by Alexander R. Povolotsky ln(3) = 4/5 +2/10*sum((1/4)^n*(1/(2*n+1)+1/(2*n+3)),n=0...infinity) [From Alexander R. Povolotsky (pevnev(AT)juno.com), Dec 18 2008]
ln(3)=sum((1/9)^(k+1)(9/(2k+1)+1/(2k+2)),k=0..infinity) [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Dec 22 2008]
Sum_{i>=1} 1/(9^i*i) + Sum_{i>=0} 1/(9^i*(i+1/2)) = 2ln3 (Huvent 2001) [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Oct 12 2009]
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EXAMPLE
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1.098612288668109691395245236922525704647490557822749451734694333637494... [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), May 16 2009]
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PROGRAM
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(PARI) { default(realprecision, 20080); x=log(3); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b002391.txt", n, " ", d)); } [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), May 16 2009]
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CROSSREFS
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Cf. A058962, A154920, A002162. [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Jan 29 2009]
Cf. A016731 Continued fraction. [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), May 16 2009]
Sequence in context: A059068 A059069 A084660 this_sequence A087044 A105415 A155920
Adjacent sequences: A002388 A002389 A002390 this_sequence A002392 A002393 A002394
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KEYWORD
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nonn,cons
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
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Fixed my PARI program, had -n Harry J. Smith (hjsmithh(AT)sbcglobal.net), May 19 2009
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