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Some periodic and non-periodic recursions. (English) Zbl 1036.11002
The authors determine all periodic recursions of the form \[ x_n= \frac{a_0+ a_1x_{n-1}+\cdots+ a_kx_{n-k}} {x_{n-k-1}}, \] where \(a_0,a_1,\dots, a_k\) are complex numbers, \(a_1,\dots, a_{k-1}\) are nonzero and \(a_k=1\). They prove that, apart from the well-known recursions \[ x_n= \frac{1}{x_{n-1}},\quad x_n= \frac{1+x_{n-1}} {x_{n-2}} \quad\text{and}\quad x_n= \frac{1+x_{n-1}+ x_{n-2}} {x_{n-3}}, \] only \[ x_n= \frac{x_{n-1}}{x_{n-2}} \quad\text{and}\quad x_n= \frac{-1-x_{n-1}+ x_{n-2}} {x_{n-3}} \] lead to periodic sequences (with periods 6 and 8).

MSC:
11B37 Recurrences
26A18 Iteration of real functions in one variable
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