Some Properties of the Multi-Figurate Numbers That Are Simultaneously Triangular and Centered Triangular

A study of the multi-figurate numbers that are simultaneously triangular and centered triangular is presented. Starting from the generalized Pell equation associated with this class of multi-figurate numbers, we describe the corresponding sequence and investigate its algebraic properties. In particular, we obtain a companion matrix representation for the sequence. In addition, we derive several identities of Tagiuri–Vajda, Catalan, Cassini, and d’Ocagne types satisfied by this sequence. These results provide a systematic description of the algebraic structure associated with the sequence of numbers that are simultaneously triangular and centered triangular.