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Combinatorial Identities For Incomplete Tribonacci Polynomials, Mark Shattuck

Applications and Applied Mathematics: An International Journal (AAM)

The incomplete tribonacci polynomials, denoted by T_n^(s)(x), generalize the usual tribonacci polynomials T_n (x) and have been shown to satisfy several algebraic identities. In this paper, we provide a combinatorial interpretation for T_n^(s)(x) in terms of weighted linear tilings involving three types of tiles. This allows one not only to supply combinatorial proofs of earlier identities for T_n^(s)(x) but also to derive new ones. In the final section, we provide a formula for the ordinary generating function of the sequence T_n^(s)(x) for a fixed s, as previously …