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%I #42 Aug 21 2023 10:19:07
%S 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,
%T 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,
%U 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
%N Decimal expansion of positive root of x^4 - x^3 - 1 = 0.
%C Also the growth constant of the Fibonacci 3-numbers A003269 [Stakhov et al.]. - _R. J. Mathar_, Nov 05 2008
%H Iain Fox, <a href="/A086106/b086106.txt">Table of n, a(n) for n = 1..20000</a>
%H Simon Baker, <a href="https://arxiv.org/abs/1711.10397">Exceptional digit frequencies and expansions in non-integer bases</a>, arXiv:1711.10397 [math.DS], 2017. See the beta(3) constant pp. 3-4.
%H A. Stakhov and B. Rozin, <a href="http://dx.doi.org/10.1016/j.chaos.2005.04.106">Theory of Binet formulas for Fibonacci and Lucas p-numbers</a>, Chaos, Solit. Fractals 27 (2006), 1162-1177.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Pisot-VijayaraghavanConstant.html">Pisot-Vijayaraghavan Constant</a>
%H <a href="/index/Al#algebraic_04">Index entries for algebraic numbers, degree 4</a>
%F 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
%e 1.380277569...
%e 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
%t RealDigits[Root[ -1 - #1^3 + #1^4 &, 2], 10, 110][[1]]
%o (PARI) polrootsreal( x^4-x^3-1)[2] \\ _Charles R Greathouse IV_, Apr 14 2014
%o (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
%Y Cf. -A230151 (other real root).
%Y Cf. A060006.
%K nonn,cons
%O 1,2
%A _Eric W. Weisstein_, Jul 09 2003