Treffer: An identity for two sequences and its combinatorial interpretation

Summary: We recall a theorem on linear recurrences that we have already proved earlier, and we use it to provide new identities. The nature of the new result allows us to combine two linear recurrences of distinct order in the identity if they satisfy some prescribed conditions about their similarity. For example, we found a rule including consecutive \(k\)- and \(\ell\)-generalized Fibonacci numbers. In addition, a combinatorial interpretation is explained if the coefficients of the recurrences are positive integers.