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Global behavior of solutions for a difference equation of third order. (English) Zbl 1369.39010
Summary: n this paper we consider the third-order rational difference equation \[ y_{n+1}\frac{y_{n-1}-y_{n-2}}{1-ay_{n-2}},n\in\mathbb N_0, \] where parameter \(a\) is a nonzero real number and the initial values \(y_{-2}\), \(y_{-1}\), \(y_0\in\mathbb R\backslash \{\frac{1}{a}\}\). We here determine both the forms and the global behavior a of the solutions of the above equation. Also, we show that the solutions are associated with Padovan numbers which contribute to explain the global behavior of the solutions.
MSC:
39A23 Periodic solutions of difference equations
11B39 Fibonacci and Lucas numbers and polynomials and generalizations
39A10 Additive difference equations
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