Netted Binomial Matrices

We prove that powers of 3-netted matrices (the entries satisfy a third-order recurrence δai,j = αai−1,j+βai−1,j−1+γai,j−1) preserve this property of nettedness, that is the entries of the e-th power satisfy δea (e) i,j = αea (e) i−1,j +βea (e) i−1,j−1+γea (e) i,j−1, where the coefficients are all instances of the same sequence, xe+1 = (β + δ)xe − (βδ + αγ)xe−1. Also, we find a matrix Tn(m) and a vector v, such that Tn(m) e ·v gives n consecutive entries of the general Fibonacci (Pell) sequence with parameter m. It generalizes the known property ( 0 1 1 1 e · (1, 0) = (Fe−1, Fe) . We close by giving a conjecture about the spectral properties of the binomial matrix we found.