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A multiset-valued Fibonacci-type sequence. (English) Zbl 1190.11010

The random Fibonacci series is defined by \[ x_0 = x_1 = 1, x_n = x_{n-1} \pm x_{n-2}, \text{ for } n \geq 2. \] In the paper, a multiset-valued recurrence is introduced by \[ f_1 = f_2 = \{1\}, f_n = (f_{n-1} \oplus f_{n-2}) \cup (f_{n-1} \ominus f_{n-2}) \] for \(n \geq 2\), where \[ X \oplus Y = \{ x + y \mid x \in X\, \&\, y \in Y \}, \]
\[ X \ominus Y = \{ x - y \mid x \in X\, \&\, y \in Y \}. \] Some properties of this recurrence are studied. For example, if \[ G(f_n) = \left | \prod_{0 \neq x \in f_n} x \right |^{\frac{1}{n|f_n|}}, \] where \(|f_n| = 2^{F_n-1}\) and \(F_n\) is \(n\)-th Fibonacci number, then \[ \lim_{n \rightarrow \infty} G(f_n)^{\frac{1}{n}} = \sqrt{\frac{1 +\sqrt{5}}{2}}. \]

MSC:

11B39 Fibonacci and Lucas numbers and polynomials and generalizations
05A15 Exact enumeration problems, generating functions
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