Turkish Journal of Analysis and Number Theory
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Turkish Journal of Analysis and Number Theory. 2020, 8(1), 1-5
DOI: 10.12691/tjant-8-1-1
Open AccessArticle

A Note on Golden Ratio and Higher Order Fibonacci Sequences

Paolo Emilio Ricci1,

1Mathematics Department, International Telematic University UniNettuno, Roma, Italia

Pub. Date: March 31, 2020

Cite this paper:
Paolo Emilio Ricci. A Note on Golden Ratio and Higher Order Fibonacci Sequences. Turkish Journal of Analysis and Number Theory. 2020; 8(1):1-5. doi: 10.12691/tjant-8-1-1

Abstract

The Lucas formula representing integer powers of the Golden ratio in terms of Fibonacci numbers is derived starting from a general result on matrix powers. The same technique is applied to the Tribonacci sequence, and the extension to higher-order Fibonacci sequences is also considered. It is shown that, by using classical results on matrix theory, the problem can be treated in a general and uniform method.

Keywords:
golden ratio Fibonacci sequence Tribonacci numbers matrix theory

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References:

[1]  G. Markowsky, The Golden Ratio – Book review, Notices of the A.M.S., 52 (3), 2005, 344-347.
 
[2]  M. Livio, The Golden Ratio: The story of Phi, the extraordinary mumber of nature, art and beauty, Broadway Books, New York, 2003.
 
[3]  G.B. Meisner, The Golden Ratio: The Divine Beauty of Mathematics, Race Point Publ., New York, 2018.
 
[4]  D. Perkins, φ, π, e & , MAAA Press, Washington D.C., 2017.
 
[5]  P. Singh, The so-called Fibonacci numbers in ancient and medieval India, Historia Math., 12, (1985), 229-244.
 
[6]  H.W. Gould, A history of the Fibonacci Qmatrix and a higher-dimensional problem, The Fibonacci Quart., 19 1981, 250-257.
 
[7]  J. Ivie, A general Q-matrix, Fibonacci Quart., 10 (1972), No. 3, 255-261, 264.
 
[8]  F.R. Gantmacher, The Theory of Matrices, Chelsea Pub. Co, New York, 1959.
 
[9]  M. Bruschi, P.E. Ricci, An explicit formulafor f() and the generating function of the generalized Lucas polynomials, Siam J.Math. Anal., 13, (1982), 162-165.
 
[10]  P.E. Ricci, Sulle potenze di una matrice, Rend. Mat. (6) 9 (1976), 179-194.
 
[11]  J.L. Brenner, Linear recurrence relations, Amer. Math. Monthly, 61 (1954), 171-173.
 
[12]  É. Lucas, Théorie des Nombres, Gauthier-Villars, Paris, 1891.
 
[13]  I.V.V. Raghavacharyulu, A.R. Tekumalla, Solution of the Difference Equations of Generalized Lucas Polynomials, J. Math. Phys., 13 (1972), 321-324.
 
[14]  R. Lidl, C. Wells, Chebyshev polynomials in several variables, J. Reine Angew. Math., 255, (1972), 104-111.
 
[15]  R. Lidl, Tschebyscheffpolynome in mehreren variabelen, J. Reine Angew. Math., 273, (1975), 178-198.
 
[16]  T.H. Koornwinder, Orthogonal polynomials in two variables which are eigenfunctions of two algebraically independent partial differential operators, I-II, Kon. Ned. Akad. Wet. Ser. A, 77, 46-66.
 
[17]  T.H. Koornwinder, Orthogonal polynomials in two variables which are eigenfunctions of two algebraically independent partial differential operators, III-IV, Indag. Math., 36. (1974), 357-381.
 
[18]  M. Bruschi, P.E. Ricci, I polinomi di Lucas e di Tchebycheff in pi`u variabili, Rend. Mat., S. VI, 13, (1980), 507-530.
 
[19]  K.B. Dunn, R. Lidl, Multi-dimensional generalizations of the Chebyshev polynomials, I-II, Proc. Japan Acad, 56 (1980), 154-165.
 
[20]  R.J. Beerends, Chebyshev polynomials in several variables and the radial part of the Laplace-Beltrami operator, Trans. Am. Math. Soc., 328, 2, 1991, 779-814.
 
[21]  H.P. Hirst, W.T. Macey, Bounding the Roots of Polynomials, The College Math. J., 28 (4) (1997), 292-295.
 
[22]  N.J.A. Sloane, S. Plouffe, The Encyclopedia of Integer Sequences, Academic Press, San Diego, 1995.
 
[23]  E.P. Miles, Jr., Generalized Fibonacci numbers and associated matrices, Amer. Math.Monthly, 67 (1960), 745-752.
 
[24]  R.A. Rosenbaum, An application of matrices to linear recursion relations, Amer. Math. Monthly, 66 (1959), 792-793.