In the present work, we explore the concept of quaternion Padovan sequences,
extending the classical Padovan sequence into the realm of quaternions.
Quaternions, introduced by Hamilton in 1843, are a number system that generalizes
complex numbers into four dimensions, with applications in various fields such as
quantum physics, computer graphics, and three-dimensional rotations. The study
begins by reviewing the definition and properties of quaternions, including their non-
commutative multiplication rules and the formulation of quaternion numbers.
The Padovan sequence is defined by the recurrence relation Pn+3 = Pn+1 + Pn with
initial conditions P0 = P1 = P2 =1. This sequence is generalized to quaternions by
defining quaternion Padovan numbers, denoted QP (a,b, c, d) , which follow a similar
recurrence relation for each coordinate of the quaternion. The study provides
detailed recurrence relations and initial conditions for these quaternion Padovan
numbers and derives several new identities and plastic-like ratio. Furthermore, the
study introduces the concept of quaternion Padovan sequences in four dimensions
and investigates their algebraic properties, including the calculation of the norm and
conjugate of these quaternion sequences.