A352748 Indices k of tribonacci numbers T(k) such that T(k+1) - (tribonacci constant)*T(k) is negative.

2, 5, 8, 11, 13, 14, 16, 17, 19, 20, 22, 23, 25, 28, 31, 34, 37, 39, 40, 42, 43, 45, 46, 48, 49, 51, 54, 57, 60, 63, 65, 66, 68, 69, 71, 72, 74, 75, 77, 80, 83, 86, 89, 91, 92, 94, 95, 97, 98, 100, 101, 103, 106, 109, 112, 115, 117, 118, 120, 121, 123, 124

Offset

1,1

Comments

The tribonacci constant, which is approximately 1.839, is described in A058265. The tribonacci constant is the only real solution to the characteristic equation (x^3 = x^2+x+1) for the tribonacci sequence. It describes the asymptotic growth of the tribonacci sequence.

The sequence doesn't contain three consecutive numbers. Also, the difference between two consecutive numbers is never more than 3.

Links

Table of n, a(n) for n=1..62.

Jeffrey Shallit, The Tribonacci constant and finite automata, arXiv preprint arXiv:2510.10834 [cs.FL], October 12 2025.

Formula

k+1 belongs to this sequence, for k>=2, if and only if (a+b*k) mod 2*Pi does not lie in (Pi/2, 3*Pi/2), where a = arg(c2*(psi-alpha)), b = arg(alpha), and T(n+1) = c1*psi^n + c2*alpha^n + c3*beta^n, where psi = 1.839... is the real zero of X^3-X^2-X-1 and alpha, beta are the complex zeros. - Jeffrey Shallit, Oct 07 2025

Example

T(5) = 4 and T(6) = 7. Therefore, T(6) - (tribonacci constant)*T(5) equals approximately -0.357147 < 0. Thus, index 5 is in this sequence.

Prog

(PARI) T(n) = ([0, 1, 0; 0, 0, 1; 1, 1, 1]^n)[1, 3]; \\ A000073
t = (1/3)*(1+(19+3*sqrt(33))^(1/3)+(19-3*sqrt(33))^(1/3)); \\ A058265
isok(k) = T(k+1) < t*T(k); \\ Michel Marcus, Apr 06 2022

Crossrefs

Complement of A352719.

Cf. A000073, A058265.

Sequence in context: A167409 A082406 A215938 * A007826 A108589 A292988

Adjacent sequences: A352745 A352746 A352747 * A352749 A352750 A352751

Keyword

nonn

Author

Tanya Khovanova and the MIT PRIMES STEP Senior group, Apr 01 2022

Extensions

More terms from Jeffrey Shallit, Oct 07 2025

Status

approved