OFFSET
0,1
COMMENTS
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)).
LINKS
Wolfdieter Lang, The Tribonacci and ABC Representations of Numbers are Equivalent, arXiv preprint arXiv:1810.09787 [math.NT], 2018.
FORMULA
b = (sqrt(3)/6)*((19 + 3*sqrt(33))^1/3 - (19 - 3*sqrt(33))^1/3).
EXAMPLE
0.606290729207199369259342197028023002949570668386421712214899686318868275...
MATHEMATICA
RealDigits[(Sqrt[3]/6) * ((19 + 3*Sqrt[33])^(1/3) - (19 - 3*Sqrt[33])^(1/3)), 10, 120][[1]] (* Amiram Eldar, May 30 2023 *)
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Wolfdieter Lang, Aug 13 2018
STATUS
approved