|
| |
|
|
A060006
|
|
Decimal expansion of real root of x^3-x-1 (sometimes called the silver constant, or the plastic constant).
|
|
24
|
|
|
|
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 all'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). [From 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]] (* From 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)); } [From 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
|
| |
|
|
