

A060006


Decimal expansion of real root of x^3  x  1 (sometimes called the silver constant, or the plastic constant).


38



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

1,2


COMMENTS

Has been also called the silver number, also the plastic number.
This is the smallest PisotVijayaraghavan 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 the paper from Witula et al.  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. 9293.
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 15th International Conference on Fibonacci Numbers and Their Applications, Eger, Hungary, 2012.


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, 13 Jan. 2015.
Simon Plouffe, Smallest PisotVijayaraghavan number to 50000 digits
Simon Plouffe, The Smallest PisotVijayaraghavan 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, PisotVijayaraghavan 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) + (1/2sqrt(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) + (1/2sqrt(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*k1/3)/(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(%) ; # 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=(xd)*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), A126772.
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



