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A060006
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Decimal expansion of real root of x^3-x-1 (sometimes called the silver constant, or the plastic constant).
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22
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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; constant; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| 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
| S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.2.2.
M. J. Gazale, Gnomon. Princeton University Press, Princeton, NJ, 1999, see Chap. VII.
D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.4, p. 236.
Ian Stewart, Tales of a neglected number, Scientific American, No. 6, 1966, pp. 92-93.
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LINKS
| 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
Ian Stewart, "Tales of a Neglected Number"
Eric Weisstein's World of Mathematics, Pisot-Vijayaraghavan Constant
Eric Weisstein's World of Mathematics, Pisot Number
Eric Weisstein's World of Mathematics, Plastic Constant
Wikipedia, Plastic number
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FORMULA
| 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
| 1.32471795724474602596090885447809734...
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MATHEMATICA
| 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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PROG
| (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
| v_2 = A001622. A072117 gives continued fraction.
Cf. A006888, A051016, A051017, A084252.
Sequence in context: A039915 A085346 A121861 * A123097 A134571 A054086
Adjacent sequences: A060003 A060004 A060005 * A060007 A060008 A060009
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KEYWORD
| cons,nice,nonn
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AUTHOR
| Fabian Rothelius (fabian.rothelius(AT)telia.com), Mar 14 2001
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EXTENSIONS
| Edited and extended by Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 03 2002
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