On the Sums of Squares of Generalized Tribonacci Numbers: Closed Formulas of Σn k=0 xkW2

Abstract

In this paper, closed forms of the sum formulas Σn k=0 xkW2 k ; Σn k=0 xk Σ Wk+1Wk and n k=0 xkWk+2Wk for the squares of generalized Tribonacci numbers are presented. As special cases, we give summation formulas of Tribonacci, Tribonacci-Lucas, Padovan, Perrin numbers and the other third order recurrence relations. We present the proofs to indicate how these formulas, in general, were discovered. Of course, all the listed formulas may be proved by induction, but that method of proof gives no clue about their discovery. Our work generalize third order recurrence relations.