Some binomial identities associated with the generalized natural number sequence

i=1 U2i−1 = U 2 n, which bears a close resemblance to ∑n i=1(2i − 1) = n . For this reason we may regard Un a generalization of the natural numbers. As in the case of Fibonacci-Lucas sequences, we define the associated sequence {Vn} as V0 = 2, V1 = p, Vn = pVn−1 − Vn−2, n ≥ 2. Of particular interest is the case of p = 3, in which we find Un = F2n and Vn = L2n. In this short note, we study two types of summations: n