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In this paper, we investigate the recurrence properties of the generalized Tribonacci sequence and present how the generalized Tribonacci sequence at negative indices can be expressed by the sequence itself at positive indices.

In this paper, closed forms of the sum formulas $\sum_{k=0}^{n}W_{k}^{2}$ \ for the squares of generalized Tetranacci numbers are presented. We also present the sum formulas $\sum_{k=0}^{n}W_{k+1}W_{k},$ $\sum_{k=0}^{n}W_{k+2}W_{k},$ and $\sum_{k=0}^{n}W_{k+3}W_{k}.$ As special cases, we give summation formulas of the of Tetranacci, Tetranacci-Lucas and some other fourth order linear recurrance sequences.

In this paper, we introduce the generalized Tetranacci quaternions and present some properties of these quaternions.