Sum of Consecutive Terms of Pell and Related Sequences

We study new identities related to the sums of adjacent terms in the Pell sequence, defined by P₍: = 2P₍-₁+P₍-₂ for n 2 and P₀=0, P₁=1, and generalize these identities for many similar sequences. We prove that the sum of N>1 consecutive Pell numbers is a fixed integer multiple of another Pell number if and only if 4 N. We consider the generalized Pell (k, i) -numbers defined by p (n): =\ 2p (n-1) +p (n-k-1) for n k+1, with p (0) =p (1) = =p (i) =0 and p (i+1) = = p (k) =1 for 0 i k-1, and prove that the sum of N=2k+2 consecutive terms is a fixed integer multiple of another term in the sequence. We also prove that for the generalized Pell (k, k-1) -numbers such a relation does not exist when N and k are odd. We give analogous results for the Fibonacci and other related second-order recursive sequences.