A060006 - OEIS
A060006
Decimal expansion of real root of x^3 - x - 1 (the plastic constant).
75
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
COMMENTS
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
Also the solution of 1/x + 1/(1+x+x^2) = 1. - Clark Kimberling, Jan 02 2020
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
REFERENCES
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.
LINKS
Brady Haran and Edmund Harriss, The Plastic Ratio, Numberphile video (2019).
Ian Stewart, Tales of a Neglected Number, Mathematical Recreations, Scientific American, Vol. 274, No. 6 (1996), pp. 102-103.
Ian Stewart, Tales of a Neglected Number, Mathematical Recreations, Scientific American, Vol. 274, No. 6 (1996), pp. 102-103.
Eric Weisstein's World of Mathematics, Pisot Number.
FORMULA
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
Equals cosh(arccosh(3*c)/3)/c, where c = sqrt(3)/2 (A010527). - Amiram Eldar, May 15 2021
EXAMPLE
1.32471795724474602596090885447809734...
MAPLE
(1/2 +sqrt(23/3)/6)^(1/3) + (1/2-sqrt(23/3)/6)^(1/3) ; evalf(%, 130) ; # R. J. Mathar, Jan 22 2013
MATHEMATICA
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 *)
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)); \\ 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
EXTENSIONS
Removed incorrect comments, Joerg Arndt, Apr 10 2016