We obtain the Binet's formula for k-Pell-Lucas numbers and as a consequence we obtain some properties for k-Pell-Lucas numbers. Also we give the generating function for the k-Pell-Lucas sequences and another expression for the general term of the sequence, using the ordinary generating function, is provided.Mathematics Subject Classification: 11B37, 05A15, 11B83.

Some identities of the k-Pell-Lucas sequence and their relationship to the Modified k-Pell sequence allow us to obtain some identities for Modified k-Pell numbers. Also the ordinary generating function for the Modified k-Pell sequence and another expression for the general term of the sequence, using the ordinary generating function, is provided.

In this paper we present the sequence of the k-Jacobsthal-Lucas numbers that generalizes the Jacobsthal-Lucas sequence introduced by Horadam in 1988. For this new sequence we establish an explicit formula for the term of order n, the well-known Binet's formula, Catalan's and d'Ocagne's Identities and a generating function. Mathematics Subject Classification 2010: 11B37, 11B83

In this paper we present some identities involving terms of k-Pell, k-Pell-Lucas and Modified k-Pell sequences. We also give some results on the column and row norms of Hankel matrices which entries are numbers of these sequences.

A proportionally modular Diophantine inequality is an expression of the form ax mod b cx , where a , b and c are positive integers. In this paper we present an algorithm that allows us to calculate the smallest positive integer that is solution of an inequality of this type. We also obtain an algorithm that computes the Frobenius number and the number of gaps of a numerical semigroup generated by three positive integers.