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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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