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Search: id:A007493
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| A007493 |
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Decimal expansion of Wallis' number, the real root of x^3 - 2*x - 5.
(Formerly M0036)
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+0
4
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| 2, 0, 9, 4, 5, 5, 1, 4, 8, 1, 5, 4, 2, 3, 2, 6, 5, 9, 1, 4, 8, 2, 3, 8, 6, 5, 4, 0, 5, 7, 9, 3, 0, 2, 9, 6, 3, 8, 5, 7, 3, 0, 6, 1, 0, 5, 6, 2, 8, 2, 3, 9, 1, 8, 0, 3, 0, 4, 1, 2, 8, 5, 2, 9, 0, 4, 5, 3, 1, 2, 1, 8, 9, 9, 8, 3, 4, 8, 3, 6, 6, 7, 1, 4, 6, 2, 6, 7, 2, 8, 1, 7, 7, 7, 1, 5, 7, 7, 5, 7, 8, 6, 0, 8, 3
(list; cons; graph; listen)
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OFFSET
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1,1
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COMMENT
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"The real solution to the equation x^3 - 2x - 5 = 0. This equation was solved by [the English mathematicaian John] Wallis [1616-1703] to illustrate Newton's method for the numerical solution of equations.
"It has since served as a test for many subsequent methods of approximation and its real root is now known to 4000 digits." [Gruenberger]
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REFERENCES
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F. Gruenberger, Computer Recreations, Scientific American, 250 (No. 4, 1984), 19-26.
W. G. Horner, A new method of solving numerical equations of all orders, by continuous approximation, Phil. Trans. Royal Soc., 1819, pp. 308-335.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
D. E. Smith, A Source Book in Mathematics, McGraw-Hill, 1929, pp. 247-248.
David Wells, "The Penguin Dictionary of Curious and Interesting Numbers," Revised Edition, Penguin Books, London, England, 1997, page 27.
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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, Wallis's Constant
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EXAMPLE
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2.094551481542326591482386540579302963857306105628239180304128529...
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MATHEMATICA
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RealDigits[ N[ 1/3* (135/2 - (3*Sqrt[1929])/2)^(1/3) + (1/2*(45 + Sqrt[1929]) )^(1/3) / 3^(2/3), 100]][[1]]
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PROGRAM
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(PARI) { default(realprecision, 20080); x=NULL; p=x^3 - 2*x - 5; rs=polroots(p); r=real(rs[1]); for (n=1, 20000, d=floor(r); r=(r-d)*10; write("b007493.txt", n, " ", d)); } [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), May 03 2009]
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CROSSREFS
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Cf. A058297 Continued fraction. [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), May 03 2009]
Sequence in context: A071120 A156649 A019693 this_sequence A136319 A152566 A021481
Adjacent sequences: A007490 A007491 A007492 this_sequence A007494 A007495 A007496
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KEYWORD
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cons,nonn
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com), Robert G. Wilson v (rgwv(AT)rgwv.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
Final digits of sequence corrected using the b-file. - N. J. A. Sloane, Aug 30 2009
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