Geodesic
Quadratic Approximation of Generalized Tribonacci Sequences
Discussiones Mathematicae. General Algebra and Applications, Tome 38 (2018) no. 2, pp. 227-237

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In this paper, we give quadratic approximation of generalized Tribonacci sequence {V_n}_n≥0 defined by V_n = rV_n−1 + sV_n−2 + tV_n−3 (n ≥ 3) and use this result to give the matrix form of the n-th power of a companion matrix of {V_n}_n≥0. Then we re-prove the cubic identity or Cassini-type formula for {V_n}_n≥0 and the Binet’s formula of the generalized Tribonacci quaternions.
Keywords: Binet’s formula, companion matrix, generalized Tribonacci sequence, Narayana number, Padovan number, quadratic approximation, Tribonacci number 
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Cerda-Morales, Gamaliel. Quadratic Approximation of Generalized Tribonacci Sequences. Discussiones Mathematicae. General Algebra and Applications, Tome 38 (2018) no. 2, pp. 227-237. http://geodesic.mathdoc.fr/item/DMGAA_2018_38_2_a5/