Padovan vectors and padovan quaternions

Padovan numbers were defined by Stewart (1996) in honor of the modern architect Richard Padovan (1935) and were first discovered in 1924 by Gerard Cordonnier. Padovan numbers are a special status of Tribonacci numbers with P0  P1  P2 1, 3 4 P  P  2 initial conditions and n n 2 n 3 P P P     general terms. The ratio between Padovan numbers is one of the important algebraic numbers because it produces plastic numbers. So much so that various studies have been carried out on both Padovan numbers and Padovan polynomial sequences until nowadays. In this study, Padovan vectors are defined for the first time by using Padovan Binet-like formula and reduction relation. Then, geometric properties of Padovan vectors such as inner product, norm and vector products are analyzed. In the last part of the study, Padovan vectors were calculated with Binet formulas in Geogebra program. In addition, the first ten Padovan numbers and Padovan vectors were calculated using the Binet forulas and shown as points and vectors in three-dimensional space. According to the Padovan vectors found, the Padovan curve was drawn in space for the first time by using the curve fitting feature of the Geogebra program. Thus, with our study, a geometric approach to Padovan number sequences was brought for the first time.