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Search: id:A060006
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| A060006 |
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Decimal expansion of real root of x^3-x-1. |
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+0
11
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| 1, 3, 2, 4, 7, 1, 7, 9, 5, 7, 2, 4, 4, 7, 4, 6, 0, 2, 5, 9, 6, 0, 9, 0, 8, 8, 5, 4, 4, 7, 8, 0, 9, 7, 3, 4, 0, 7, 3, 4, 4, 0, 4, 0, 5, 6, 9, 0, 1, 7, 3, 3, 3, 6, 4, 5, 3, 4, 0, 1, 5, 0, 5, 0, 3, 0, 2, 8, 2, 7, 8, 5, 1, 2, 4, 5, 5, 4, 7, 5, 9, 4, 0, 5, 4, 6, 9, 9, 3, 4, 7, 9, 8, 1, 7, 8, 7, 2, 8, 0, 3, 2, 9, 9, 1
(list; cons; graph; listen)
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OFFSET
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1,2
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COMMENT
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Has been also called the silver number, also the plastic number.
This is the smallest Pisot-Vijayaraghavan number, v_3. In general v_n is the smallest positive real solution to the equation (v_n)^n = v_n + 1.
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REFERENCES
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S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.2.2.
M. J. Gazale, Gnomon. Princeton University Press, Princeton, NJ, 1999, see Chap. VII.
Ian Stewart, Tales of a neglected number, Scientific American, No. 6, 1966, pp. 92-93.
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LINKS
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Harry J. Smith, Table of n, a(n) for n=1,...,20000
S. Plouffe, Smallest Pisot-Vijayaraghavan number to 50000 digits
S. Plouffe, The Smallest Pisot-Vijayaraghavan number
F. Rothelius, Formulae
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics
Eric Weisstein's World of Mathematics, Pisot Number
Eric Weisstein's World of Mathematics, Plastic Constant
Wikipedia, Plastic number
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FORMULA
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Another formula: ((1/2)+((1/6)*sqrt(23/3)))^(1/3) + ((1/2)-((1/6)*sqrt(23/3)))^(1/3) [From Eric Desbiaux (moongerms(AT)wanadoo.fr), Oct 17 2008]
(1/2+sqrt(23/108))^(1/3) + (1/2-sqrt(23/108))^(1/3) - Henry Bottomley (se16(AT)btinternet.com), May 22 2003
CubeRoot(1 + CubeRoot(1 + CubeRoot(1 + CubeRoot(1 + ...)))) - Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), Nov 26 2004
sqrt(1+1/sqrt(1+1/sqrt(1+1/sqrt(1+...)))) - Gerald McGarvey (gerald.mcgarvey(AT)comcast.net), Mar 18 2006
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EXAMPLE
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1.32471795724474602596090885447809734...
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MATHEMATICA
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RealDigits[ Solve[x^3 - x - 1 == 0, x][[1, 1, 2]], 10, 111][[1]] [From Robert G. Wilson v (rgwv(AT)rgwv.com), Sep 30 2009]
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PROGRAM
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(PARI) { allocatemem(932245000); default(realprecision, 20080); x=solve(x=1, 2, x^3 - x - 1); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b060006.txt", n, " ", d)); } [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), Jul 01 2009]
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CROSSREFS
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v_2 = A001622. A072117 gives continued fraction.
Cf. A006888, A051016, A051017, A084252.
Sequence in context: A039915 A085346 A121861 this_sequence A123097 A134571 A054086
Adjacent sequences: A060003 A060004 A060005 this_sequence A060007 A060008 A060009
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KEYWORD
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cons,nice,nonn
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AUTHOR
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Fabian Rothelius (fabian.rothelius(AT)telia.com), Mar 14 2001
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EXTENSIONS
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Edited and extended by Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 03 2002
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