|
%I #16 May 30 2023 02:22:46
%S 6,0,6,2,9,0,7,2,9,2,0,7,1,9,9,3,6,9,2,5,9,3,4,2,1,9,7,0,2,8,0,2,3,0,
%T 0,2,9,4,9,5,7,0,6,6,8,3,8,6,4,2,1,7,1,2,2,1,4,8,9,9,6,8,6,3,1,8,8,6,
%U 8,2,7,5,2,8,1,1,4,5,6,6,2,0,3,1,3,2,7,9,3,0,3,7,9,4,0,2,3,4,0,9,8,2,9
%N Decimal expansion of the absolute value of the imaginary part of the complex conjugated solutions of the tribonacci equation t^3 - t^2 - t - 1 = 0.
%C The three solution of the eigenvalues of the transfer matrix (Q matrix) of the tribonacci recurrence A000073 Q = matrix[[1, 1, 1], [1, 0, 0], [0, 1, 0]], that is, the three solutions of t^3 - t^2 - t - 1 = 0 are: t = (1 + (19 + 3*sqrt(33))^(1/3) + (19 - 3*sqrt(33))^(1/3))/3 = A058265 (the real tribonacci constant) and the complex conjugated solutions (a + b*i) and (a - b*i) with a = -(t - 1)/2 and b = (sqrt(3)/6)*((19 + 3*sqrt(33))^(1/3) - (19 - 3*sqrt(33))^(1/3)).
%H Wolfdieter Lang, <a href="https://arxiv.org/abs/1810.09787">The Tribonacci and ABC Representations of Numbers are Equivalent</a>, arXiv preprint arXiv:1810.09787 [math.NT], 2018.
%F b = (sqrt(3)/6)*((19 + 3*sqrt(33))^1/3 - (19 - 3*sqrt(33))^1/3).
%e 0.606290729207199369259342197028023002949570668386421712214899686318868275...
%t RealDigits[(Sqrt[3]/6) * ((19 + 3*Sqrt[33])^(1/3) - (19 - 3*Sqrt[33])^(1/3)), 10, 120][[1]] (* _Amiram Eldar_, May 30 2023 *)
%Y Cf. A000073, A058265.
%K nonn,cons
%O 0,1
%A _Wolfdieter Lang_, Aug 13 2018
|
