On generalized (k, r)-Pell and (k, r)-Pell–Lucas numbers

We introduce new kinds of k-Pell and k-Pell–Lucas numbers related to the distance between numbers by a recurrence relation and show their relation to the (k,r)-Pell and (k,r)-Pell–Lucas numbers. These sequences differ both according to the value of the natural number k and the value of a new parameter r in the definition of this distance. We give several properties of these sequences. In addition, we establish the generating functions, some important identities, as well as the sum of the terms of the generalized (k,r)-Pell and (k,r)-Pell–Lucas numbers. Furthermore, we indicate another way to obtain the generalized (k,r)-Pell and (k,r)-Pell–Lucas sequences from the generating function, in connection to graphs.