The incomplete tribonacci polynomials, denoted by Tn(s)(x), generalize the usual tribonacci polynomials Tn (x) and have been shown to satisfy several algebraic identities. In this paper, we provide a combinatorial interpretation for Tn(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 Tn(s)(x) but also to derive new ones. In the final section, we provide a formula for the ordinary generating function of the sequence Tn(s)(x) for a fixed s, as previously requested. Our derivation is combinatorial in nature and makes use of an identity relating Tn(s)(x) to Tn (x).
Combinatorial Identities for Incomplete Tribonacci Polynomials,
Applications and Applied Mathematics: An International Journal (AAM), Vol. 10,
1, Article 3.
Available at: https://digitalcommons.pvamu.edu/aam/vol10/iss1/3