Abstract
We consider the sequence {T n } ∞0 defined by
, with initial conditions
, where the a r and c r are integers and a0 ≠ 0. (If the c r are all zero, {T n }∞ 0 becomes the null sequence. In this case Theorems 1 and 2 below are trivial.) In (1) m ≥ 0 is a fixed integer. We referee to (1) as an (m+1)th order recurrence relation or an (m+1)th order difference equation. Thus {T n } is an integer sequence. The purpose of our present paper is to generalize results which we obtained [2] for a sequence {T n } defined by a second order recurrence relation (m = 1 in (1)), the Fibonacci and Lucas sequences being important special cases. (The case m = 0 is trivial.)
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References
Freitag, H. T. ”A Property of Unit Digits of Fibonacci Numbers and a ’Non-Elegant’ Way of Expressing Gn.” Proceedings of the First International Conference on Fibonacci Numbers and Their Applications, University of Patras, Greece, August 27–31, 1984.
Freitag, H. T. and Phillips, G. M. ”A Congruence Relation for Certain Recursive Sequences.” Fibonacci Quarterly 24, (1986): pp 332–335.
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© 1988 Springer Science+Business Media New York
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Freitag, H.T., Phillips, G.M. (1988). A Congruence Relation for a Linear Recursive Sequence of Arbitrary Order. In: Philippou, A.N., Horadam, A.F., Bergum, G.E. (eds) Applications of Fibonacci Numbers. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-7801-1_5
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DOI: https://doi.org/10.1007/978-94-015-7801-1_5
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-8447-7
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