1. Introduction. Our motivation for this research is to describe an iterative sequence of n-dimensional points mathematically. Consider the famous Fibonacci sequence 1,2,3, 5,8,13,... with the relationship Fn = Fn-1 + Fn-2 for n > 1 and Fo = F1 = 1. By examination, we notice that every Fibonacci number Fn (n > 2) can be obtained from the previous ones iteratively. What about a set of n-dimensional points in the vector space Rn? How do we use a mathematical function to trace the points? For example, a set of 3-dimensional points is given by Pi = (1,2,3),P2 = (1,3,5), P3 = (2,5,8), P4 = (3,8,13), P5 = (5,13,21),.... An explicit formula for the points is given by Pk = (Fk, Fk+2, Fk+3) for i > 1. Similar to the Fibonacci sequence, we may describe this sequence of points by the relationship Pk = Pk-1+Pk-2 for k > 2 and the initial conditions P1 = (1,2,3),P2 = (1,3,5). Another approach is to use a 3-tuple linear function f = (χ2-χ1,2χ2-χ1,3χ2-χ1) to describe the sequence. One can easily check that f(1,2,3) = (1,3,5), f(1,3,5) = (2,5,8), f(2,5,8) = (3,8,13), and f(3,8,13) = (5,13,21). Iteratively, f(Pk) = Pk+1 for all k > 1. A natural question is, if S = {Ai,A2,..., Am+1} with Ai = (ai1,ai2,...,ain) ∈ Rn, can we find a multivariable function f: Rn →Rn such that f(Ai) = Ai+1 for all i = 1,2,..., m? If so can we construct a function satisfying certain constraints? In this paper, we focus on the construction of such tracing functions.
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