A086106 Decimal expansion of positive root of x^4 - x^3 - 1 = 0.

1, 3, 8, 0, 2, 7, 7, 5, 6, 9, 0, 9, 7, 6, 1, 4, 1, 1, 5, 6, 7, 3, 3, 0, 1, 6, 9, 1, 8, 2, 2, 7, 3, 1, 8, 7, 7, 8, 1, 6, 6, 2, 6, 7, 0, 1, 5, 5, 8, 7, 6, 3, 0, 2, 5, 4, 1, 1, 7, 7, 1, 3, 3, 1, 2, 1, 1, 2, 4, 9, 5, 7, 4, 1, 1, 8, 6, 4, 1, 5, 2, 6, 1, 8, 7, 8, 6, 4, 5, 6, 8, 2, 4, 9, 0, 3, 5, 5, 0, 9, 3, 7

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-Vijayaraghavan Constant

Index entries for algebraic numbers, degree 4

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

Cf. -A230151 (other real root).

Cf. A060006.

Sequence in context: A155979 A120669 A021267 * A199731 A010626 A370413

Adjacent sequences: A086103 A086104 A086105 * A086107 A086108 A086109

Keyword

nonn,cons

Author

Eric W. Weisstein, Jul 09 2003

Status

approved