Sequences related to the Pell generalized equation

We consider sequences of the type An = 6An−1 − An−2, A0 = r, A1 = s (r and s integers) and show that all sequences that solve particular cases of the Pell generalized equation are expressible as a constant times one of four particular sequences of the same type. Let α = 3 + 2 √ 2, β = 3 − 2 √ 2 be the roots of the polynomial x − 6x+ 1. Note that α+β = 6, αβ = 1, α−β = 4 √ 2. Also let γ = 1+ √ 2, δ = 1− √ 2. Then γ = α, δ = β, γδ = −1. We take γ = α 1 2 , δ = −β 1 2 . Consider the sequence An defined by An = 6An−1 −An−2, A0 = r, A1 = s, (1) where r and s are integers. The object of this contribution is to show that all sequences of the type given by Equation 1 that solve particular cases of the Pell generalized equation (see [3]) are expressible as a constant times one of four particular sequences of the same type. The generating function of An is given by g(x) = r + (s− 7r)x+ (6r − s)x (1− x)(1− 6x+ x) , from which we get the closed form An = 2sγα− 2rγ 8 √ 2γ α − 2rγβ + 2sδβ 8 √ 2δ .