A316710 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.

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, 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, 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

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

Table of n, a(n) for n=0..102.

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

Cf. A000073, A058265.

Sequence in context: A161419 A136526 A097715 * A198499 A092605 A180318

Adjacent sequences: A316707 A316708 A316709 * A316711 A316712 A316713

Keyword

nonn,cons

Author

Wolfdieter Lang, Aug 13 2018

Status

approved