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In this paper, we consider a generalization of Horadam sequence {w n } which is defined by the recurrence relation w n = χ (n)w n-1 + cw n-2 , where χ (n) = a if n is even, χ (n) = b if n is odd with arbitrary initial conditions w 0 , w 1 and nonzero real numbers a, b and c. As a special case, by taking the initial conditions 0, 1 and 2, b we define the sequences {u n } and {v n }, respectively. The main purpose of this study is to derive some basic properties of the sequences {u n }, {v n } and {w n } by using a matrix approach.

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Abstract. In this paper, we consider the Fibonacci conditional sequence ff n g and the Lucas conditional sequence fl n g. We derive some properties of Fibonacci and Lucas conditional sequences by using the matrix method. Our results are elegant as the results for ordinary Fibonacci and Lucas sequences.

In this short paper we establish identities involving sums of products of binomial coefficients and coefficients that satisfy the general second-order linear recurrence. We obtain generalizations of identities of Carlitz, Prodinger and Haukkanen.