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

Voir la notice de l'article provenant de la source Library of Science

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 
@article{DMGAA_2018_38_2_a5,
     author = {Cerda-Morales, Gamaliel},
     title = {Quadratic {Approximation} of {Generalized} {Tribonacci} {Sequences}},
     journal = {Discussiones Mathematicae. General Algebra and Applications},
     pages = {227--237},
     publisher = {mathdoc},
     volume = {38},
     number = {2},
     year = {2018},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DMGAA_2018_38_2_a5/}
}
TY  - JOUR
AU  - Cerda-Morales, Gamaliel
TI  - Quadratic Approximation of Generalized Tribonacci Sequences
JO  - Discussiones Mathematicae. General Algebra and Applications
PY  - 2018
SP  - 227
EP  - 237
VL  - 38
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DMGAA_2018_38_2_a5/
LA  - en
ID  - DMGAA_2018_38_2_a5
ER  - 
%0 Journal Article
%A Cerda-Morales, Gamaliel
%T Quadratic Approximation of Generalized Tribonacci Sequences
%J Discussiones Mathematicae. General Algebra and Applications
%D 2018
%P 227-237
%V 38
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DMGAA_2018_38_2_a5/
%G en
%F DMGAA_2018_38_2_a5
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/

[1] G. Cerda-Morales, On a Generalization of Tribonacci Quaternions, Mediterranean J. Math. 14:239 (2017) 1–12. doi:10.1007/s00009-017-1042-3

[2] M. Elia, Derived sequences, the Tribonacci recurrence and cubic forms, The Fibonacci Quarterly 39 (2001) 107–109.

[3] M. Feinberg, Fibonacci-Tribonacci, The Fibonacci Quarterly 1 (1963) 71–74.

[4] W. Gerdes, Generalized Tribonacci numbers and their convergent sequences, The Fibonacci Quarterly 16 (1978) 269–275.

[5] E. Kiliç, Tribonacci sequences with certain indices and their sums, Ars Comb. 86 (2008) 13–22.

[6] T. Koshy, Fibonacci and Lucas Numbers With Applications (John Wiley & Sons, INC, 2001).

[7] K. Kuhapatanakul and L. Sukruan, The generalized Tribonacci numbers with negative subscripts, Integers 14, Paper A32, 6 p. (2014).

[8] S. Pethe, Some identities for Tribonacci sequences, The Fibonacci Quarterly 26 (1988) 144–151.

[9] J.L. Ramírez and V.F. Sirvent, A note on the k-Narayana sequence, Ann. Math. Inform. 45 (2015) 91–105.

[10] A.G. Shannon and A.F. Horadam, Some properties of third-order recurrence relations, The Fibonacci Quarterly 10 (1972) 135–146.

[11] W.R. Spickerman, Binet’s formula for the Tribonacci numbers, The Fibonacci Quarterly 20 (1982) 118–120.

[12] M.E. Waddill and L. Sacks, Another generalized Fibonacci sequence, The Fibonacci Quarterly 5 (1967) 209–222.

[13] C.C. Yalavigi, Properties of Tribonacci numbers, The Fibonacci Quarterly 10 (1972) 231–246.