Some identities for compositions into parts of size at most 3

In this paper, we consider compositions of integers when only parts of size at most 3 are allowed in both composition and its conjugate composition. First we obtain a relation between the number of such compositions and the Fibonacci numbers. Then we provide combinatorial identities between these compositions and compositions into 1’s and 2’s, compositions into odd parts, and compositions into parts greater than 1. Further, several generalized results for these compositions are obtained.