the coefficients, the problem becomes much more difficult. Even in case of the usual Fibonacci system, no compact formula is known for the so-called Fibagonci sequence $(B(n))_{n \in \mathbb{N}}$ counting the number of representations of $n$ as sum of (possibly repeating) Fibonacci numbers.
One can also ask the question about numeration systems which allow coefficients ≥ 2 even in the greedy expansion of an integer. An example of these is the Ostrowski numeration system based on sequences defined by linear recurrences of second order with non-constant coefficients. Such numeration systems have been considered by Berstel [1] who shows that a formula similar to (5) is valid for counting the number of representations of $n$. Other properties of these numeration systems are to be explored.
Acknowledgements
The authors acknowledge partial support by Czech Science Foundation GA ČR 201/05/0169, and by the grant LC06002 of the Ministry of Education, Youth, and Sports of the Czech Republic.
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