  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\geq 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 $n\geq k+1$, with
$p(0)=p(1)=\cdots =p(i)=0$ and $p(i+1)=\cdots = p(k)=1$ for $0\leq i\leq k-1$,
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.
