In this paper, the authors have proposed an innovative approach for generating a pseudo random ternary sequence by using a primitive polynomial, trace function, and Legendre symbol over odd characteristics field. Let p be an odd prime number, F_P be an odd characteristic prime field, and m be the degree of the primitive polynomial f(x) Let w be its zero and a primitive element in F_pm* In the beginning, a primitive polynomial f(x) generates maximum length vector sequence, then the trace function Tr(.) is used to map an element of the extension field (F_pm) to an element of the prime field F_p then non-zero scalar A∈F_p is added to the trace value, and finally the Legendre symbol (a/p) is utilized to map the scalars into ternary sequence having the values, {0,1,and -1} By applying the new parameter A the period of the sequence is extended to its maximum value that is n=p^m-1 Hence, our proposed sequence has some parameters such as p,m,and A This paper mathematically explains the properties of the proposed ternary sequence such as period and autocorrelation. Additionally, these properties are also justified based on some experimental results.