In this paper, we derive expressions for the sums of first four powers of balancing and Lucas-balancing numbers by using the telescoping summation formula. Further, we use these new results to obtain other closed form expressions studied earlier.
A generalization of the well-known Lucas sequence is the k -Lucas sequence with some fixed integer k ≥ 2 .The first k terms of this sequence are 0, . . . , 0, 2, 1 , and each term afterwards is the sum of the preceding k terms. In this paper, we determine all repdigits, which are expressible as sums of two k -Lucas numbers. This work generalizes a prior result of Şiar and Keskin who dealt with the above problem for the particular case of Lucas numbers and a result of Bravo and Luca who searched for repdigits that are k -Lucas numbers.
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