Identities for Tribonacci-related sequences

We establish some identities relating two sequences that are, as explained, related to the Tribonacci sequence. One of these sequences bears the same resemblance to the Tribonacci sequence as the Lucas sequence does to the Fibonacci sequence. Defining a matrix that we call Tribomatrix, which extends the Fibonacci matrix, we see that the other sequence is related to the sum of the determinants of the 2nd order principal minors of this matrix. 1 Antefacts Let Sn be the generalized Lucas sequence, also called generalized Tribonacci sequence, that is Sn+1 = Sn + Sn−1 + Sn−2, S0 = 3, S1 = 1, S2 = 3. Sn is sequence A001644 in [2]. Let {α, β, γ} be the roots of the characteristics polynomial x3 −x2 −x− 1 = 0 (for an explicit expression see [1]). Let us assume that α is the real root, β and γ are the complex conjugate roots. We have α = 1.8392286..., |β| = |γ| = 0.737353... (see [3]). The Binet’s formula (see [1]) is