Sum Formulas for Generalized Fifth-Order Linear Recurrence Sequences
Yüksel Soykan *
Department of Mathematics, Faculty of Art and Science, Zonguldak Bülent Ecevit University, 67100, Zonguldak, Turkey.
*Author to whom correspondence should be addressed.
Abstract
In this paper, closed forms of the summation formulas for generalized Pentanacci numbers are presented. Then, some previous results are recovered as particular cases of the present results. As special cases, we give summation formulas of Pentanacci, Pentanacci-Lucas, fifth order Pell, fifth order Pell-Lucas, fifth order Jacobsthal and fifth order Jacobsthal-Lucas sequences. We present the proofs to indicate how these formulas, in general, were discovered. In fact, all the listed formulas of the special cases of of the main theorems may be proved by induction, but that method of proof gives no clue about their discovery.
Keywords: Pentanacci numbers, Pentanacci-Lucas numbers, sum formulas, summing formulas