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A060006 Decimal expansion of real root of x^3 - x - 1 (sometimes called the silver constant, or the plastic constant). 53
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; text; internal format)
OFFSET

1,2

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

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.

Dom Hans van der Laan, Le nombre plastique: Quinze leçons sur l’ordonnance architectonique, Brill Academic Pub., Leiden 1960.

Ian Stewart, Tales of a Neglected Number, Scientific American, No. 6, June 1996, pp. 92-93.

Ian Stewart, A Guide to Computer Dating (Feedback), Scientific American, Vol. 275 No. 5, November 1996, p. 118.

LINKS

Harry J. Smith, Table of n, a(n) for n = 1..20000

Alex Bellos, The golden ratio has spawned a beautiful new curve: the Harriss spiral, The Guardian, Jan 13 2015.

Simon Plouffe, Smallest Pisot-Vijayaraghavan number to 50000 digits

Simon Plouffe, The Smallest Pisot-Vijayaraghavan number

F. Rothelius, Formulae

Ian Stewart, Tales of a Neglected Number

M. Waldschmidt, Lectures on Multiple Zeta Values, IMSC 2011.

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

FORMULA

(1/2+sqrt(23/108))^(1/3) + (1/2-sqrt(23/108))^(1/3). - Henry Bottomley, May 22 2003

CubeRoot(1 + CubeRoot(1 + CubeRoot(1 + CubeRoot(1 + ...)))). - Gerald McGarvey, Nov 26 2004

sqrt(1+1/sqrt(1+1/sqrt(1+1/sqrt(1+...)))). - Gerald McGarvey, Mar 18 2006

(1/2 +sqrt(23/3)/6)^(1/3) + (1/2-sqrt(23/3)/6)^(1/3). - Eric Desbiaux, Oct 17 2008

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

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) polrootsreal(x^3-x-1)[1] \\ Charles R Greathouse IV, Aug 28 2016

CROSSREFS

Cf. A001622. A072117 gives continued fraction.

Cf. A006888, A051016, A051017, A084252, A075778 (inverse), A126772.

Other Pisot numbers: A086106, A092526, A228777, A293506, A293508, A293509, A293557.

Sequence in context: A085346 A121861 A317736 * A123097 A209706 A134571

Adjacent sequences:  A060003 A060004 A060005 * A060007 A060008 A060009

KEYWORD

cons,nice,nonn

AUTHOR

Fabian Rothelius, Mar 14 2001

EXTENSIONS

Edited and extended by Robert G. Wilson v, Aug 03 2002

Removed incorrect comments, Joerg Arndt, Apr 10 2016

STATUS

approved

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Last modified December 12 00:07 EST 2018. Contains 318052 sequences. (Running on oeis4.)