Generalized Lucas numbers of the form 3 × 2^m

For an integer $k\geq 2$, let $(L_n^{(k)})_n$ be the k-generalized Lucas sequence which starts with $0,\ldots,0,2,1$ (k terms) and each term afterwards is the sum of the k preceding terms. In this paper, we look the k-generalized Lucas numbers of the form $3\times 2^m$ i.e. we study the Diophantine equation $L^{(k)}_n = 3\times 2^m$ in positive integers n, k, m with $k \geq 2$.

Convolutions of the generalized Jacobsthal and generalized Lucas numbers

In this paper we consider the generalized Jacobsthal Jn,m and the generalized Jacobsthal-Lucas numbers jn,m. Also, we introduce new sequences of numbers An,m, Bn,m, Cn,m and Dn,m. Namely, these new sequences are convolutions of the sequences Jn,m and jn,m. Further, we find the generating functions and some recurrence relations for these sequences of numbers.

Some properties of the generalized Fibonacci and Lucas numbers

In this paper we consider the generalized Fibonacci numbers Fn,m and the generalized Lucas numbers Ln,m. Also, we introduce new sequences of numbers An,m, Bn,m, Cn,m and Dn,m. Further, we find the generating functions and some recurrence relations for these sequences of numbers.