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A060006 Decimal expansion of real root of x^3-x-1 (sometimes called the silver constant, or the plastic constant). 35
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, v_3. In general v_n is the smallest positive real solution to the equation (v_n)^n = v_n + 1.

The decomposition of the polynomial x^3 - x - 1 in comments to A218197 is presented. See also Witula et al's paper. - Roman Witula, Oct 22 2012

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.

M. Waldschmidt, Lectures on Multiple Zeta Values (IMSC 2011), http://www.math.jussieu.fr/~miw/articles/pdf/MZV2011IMSc.pdf

R. Witula, E. Hetmaniok, D. Slota, Sums of the powers of any order roots taken from the roots of a given polynomial, submitted to Proceedings of the 15'th International Conference on Fibonacci Numbers and Their Applications, Eger, Hungary, 2012.

LINKS

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

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"

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

EXAMPLE

1.32471795724474602596090885447809734...

MAPLE

(1/2 +sqrt(23/3)/6)^(1/3) + (1/2-*sqrt(23/3)/6)^(1/3) ; evalf(%) ; # 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 *)

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]

CROSSREFS

v_2 = A001622. A072117 gives continued fraction.

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

Sequence in context: A039915 A085346 A121861 * 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

STATUS

approved

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Last modified November 21 04:39 EST 2014. Contains 249769 sequences.