A060006 revision #191
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A060006
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Decimal expansion of real root of x^3 - x - 1 (the plastic constant).
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75
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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
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OFFSET
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1,2
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COMMENTS
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Has been also called the silver number, also the plastic number.
This is the smallest Pisot-Vijayaraghavan number.
The name "plastic number" goes back to the Dutch Benedictine monk and architect Dom Hans van der Laan, who gave this name 4 years after the discovery of the number by the French engineer Gérard Cordonnier in 1924, who used the name "radiant number". - Hugo Pfoertner, Oct 07 2018
Sometimes denoted by the symbol rho. - Ed Pegg Jr, Feb 01 2019
Given any complex p such that real(p)>-1, this constant is the only real solution of the equation z^p+z^(p+1)=z^(p+3), and the only attractor of the complex mapping z->M(z,p), where M(z,p)=(z^p+z^(p+1))^(1/(p+3)), convergent from any complex plane point. - Stanislav Sykora, Oct 14 2021
The Pisot-Vijayaraghavan numbers were named after the French mathematician Charles Pisot (1910-1984) and the Indian mathematician Tirukkannapuram Vijayaraghavan (1902-1955). - Amiram Eldar, Apr 02 2022
The sequence a(n) = v_3^floor(n^2/4) where v_n is the smallest, positive, real solution to the equation (v_n)^n = v_n + 1 satisfies the Somos-5 recursion a(n+3)*a(n-2) = a(n+2)*a(n-1) + a(n+1)*a(n) for all n in Z. Also true if floor is removed. - Michael Somos, Mar 24 2023
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REFERENCES
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Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.2.2.
Midhat J. Gazalé, Gnomon: From Pharaohs to Fractals, Princeton University Press, Princeton, NJ, 1999, see Chap. VII.
Donald E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.4, p. 236.
Ian Stewart, A Guide to Computer Dating (Feedback), Scientific American, Vol. 275 No. 5, November 1996, p. 118.
Dom Hans van der Laan, Le nombre plastique: Quinze leçons sur l’ordonnance architectonique, Brill Academic Pub., Leiden, 1960.
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LINKS
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FORMULA
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Equals (1/2+sqrt(23/108))^(1/3) + (1/2-sqrt(23/108))^(1/3). - Henry Bottomley, May 22 2003
Equals CubeRoot(1 + CubeRoot(1 + CubeRoot(1 + CubeRoot(1 + ...)))). - Gerald McGarvey, Nov 26 2004
Equals sqrt(1+1/sqrt(1+1/sqrt(1+1/sqrt(1+...)))). - Gerald McGarvey, Mar 18 2006
Equals (1/2 +sqrt(23/3)/6)^(1/3) + (1/2-sqrt(23/3)/6)^(1/3). - Eric Desbiaux, Oct 17 2008
Equals Sum_{k >= 0} 27^(-k)/k!*(Gamma(2*k+1/3)/(9*Gamma(k+4/3)) - Gamma(2*k-1/3)/(3*Gamma(k+2/3))). - Robert Israel, Jan 13 2015
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EXAMPLE
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1.32471795724474602596090885447809734...
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MAPLE
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(1/2 +sqrt(23/3)/6)^(1/3) + (1/2-sqrt(23/3)/6)^(1/3) ; evalf(%, 130) ; # R. J. Mathar, Jan 22 2013
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MATHEMATICA
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RealDigits[ Solve[x^3 - x - 1 == 0, x][[1, 1, 2]], 10, 111][[1]] (* Robert G. Wilson v, Sep 30 2009 *)
s = Sqrt[23/108]; RealDigits[(1/2 + s)^(1/3) + (1/2 - s)^(1/3), 10, 111][[1]] (* Robert G. Wilson v, Dec 12 2017 *)
RealDigits[Root[x^3-x-1, 1], 10, 120][[1]] (* or *) RealDigits[(Surd[9-Sqrt[69], 3]+Surd[9+Sqrt[69], 3])/(Surd[2, 3]Surd[9, 3]), 10, 120][[1]] (* Harvey P. Dale, Sep 04 2018 *)
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PROG
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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)); \\ Harry J. Smith, Jul 01 2009
(PARI) (1/2 +sqrt(23/3)/6)^(1/3) + (1/2-sqrt(23/3)/6)^(1/3) \\ Altug Alkan, Apr 10 2016
(PARI) default(realprecision, 110); digits(floor(solve(x=1, 2, x^3 - x - 1)*10^105)) /* Michael Somos, Mar 24 2023 */
(Magma) SetDefaultRealField(RealField(100)); ((3+Sqrt(23/3))/6)^(1/3) + ((3-Sqrt(23/3))/6)^(1/3); // G. C. Greubel, Mar 15 2019
(Sage) numerical_approx(((3+sqrt(23/3))/6)^(1/3) + ((3-sqrt(23/3))/6)^(1/3), digits=100) # G. C. Greubel, Mar 15 2019
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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proposed
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