OFFSET
1,2
COMMENTS
Also the growth constant of the Fibonacci 3-numbers A003269 [Stakhov et al.]. - R. J. Mathar, Nov 05 2008
LINKS
Iain Fox, Table of n, a(n) for n = 1..20000
Simon Baker, Exceptional digit frequencies and expansions in non-integer bases, arXiv:1711.10397 [math.DS], 2017. See the beta(3) constant pp. 3-4.
A. Stakhov and B. Rozin, Theory of Binet formulas for Fibonacci and Lucas p-numbers, Chaos, Solit. Fractals 27 (2006), 1162-1177.
Eric Weisstein's World of Mathematics, Pisot Number.
FORMULA
Equals (1 + (A^2 + sqrt(A^4 - 16*u*A^2 + 2*A))/A)/4 with A = sqrt(8*u + 3/2), u = (-(Bp/2)^(1/3) + (Bm/2)^(1/3)*(1 - sqrt(3)*i)/2 - 3/8)/6, with Bp = 27 + 3*sqrt(3*283), Bm = 27 - 3*sqrt(3*283), and i = sqrt(-1). (Standard computation of a quartic.) The other (negative) real root -A230151 is obtained by using in the first formula the negative square root. The other two complex roots are obtained by replacing A by -A in these two formulas. - Wolfdieter Lang, Aug 19 2022
EXAMPLE
1.380277569...
The four solutions are the present one, -A230151, and the two complex ones 0.2194474721... - 0.9144736629...*i and its complex conjugate. - Wolfdieter Lang, Aug 19 2022
MATHEMATICA
RealDigits[Root[ -1 - #1^3 + #1^4 &, 2], 10, 110][[1]]
PROG
(PARI) polrootsreal( x^4-x^3-1)[2] \\ Charles R Greathouse IV, Apr 14 2014
(PARI) default(realprecision, 20080); x=solve(x=1, 2, x^4 - x^3 - 1); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b086106.txt", n, " ", d)); \\ Iain Fox, Oct 23 2017
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Eric W. Weisstein, Jul 09 2003
STATUS
approved