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 $P_n:=2P_{n-1}+P_{n-2}$ for $n\ge 2$ and $P_0=0,P_1=1$, 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 $4\mid N$. We consider the generalized Pell $(k,i)$-numbers defined by $p(n):=2p(n-1)+p(n-k-1)$ for $n\ge k+1$, with $p(0)=p(1)=\cdots =p(i)=0$ and $p(i+1)=\cdots =p(k)=1$ for $0\le i\le k-1$, 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.