A Study of the Symmetry of the Tricomplex Repunit Sequence with Repunit Sequence

In this paper, we introduce a new family of sequences related to Horadam-type sequences. Specifically, we consider the repunit sequence {rn}n≥0, which is defined by the initial terms r0=0 and r1=1 and follows the Horadam recurrence relation given by rn=11rn−1−10rn−2 for n≥2. Many studies have explored generalizations of integer sequences in different directions: some by preserving the initial terms, some by preserving the recurrence relation, and some by considering different numerical sets beyond positive integers. In this article, we take the third approach. Specifically, we study these sequences in the context of the tricomplex ring T. We define the Tricomplex Repunit sequence {trn}n≥0, with initial terms tr0=(0,1,11) and tr1=(1,11,111), and governed by the recurrence relation trn=11trn−1−10trn−2, for n≥2. This sequence is also a Horadam-type sequence but defined in the tricomplex ring T. In this paper, we establish the properties of the Tricomplex Repunit sequence and establish several new as well as well-known identities associated with it, including Binet’s formula, Tagiuri–Vajda’s identity, d’Ocagne’s identity, and Catalan’s identity. We also derive the generating function for this sequence. Furthermore, we study various additional properties of these generalized sequences and establish results concerning the summation of terms related to the Tricomplex Repunit sequence, and one of our main goals is to determine analogous or symmetrical properties for the Tricomplex Repunit sequence to those we know for the ordinary repunit sequence.