Abstract
We investigate the conditions on an integer sequence f ( n ) , n ∈ N , with f ( 1 ) = 0 , such that the sequence q ( n ) , computed recursively via q ( n ) = q ( n − q ( n − 1 ) ) + f ( n ) , with q ( 1 ) = 1 , exists. We prove that f ( n ) is ‘slow’, that is, f ( n + 1 ) − f ( n ) ∈ { 0 , 1 } , n ≥ 1 , is a sufficient but not necessary condition for the existence of sequence q. Sequences q defined in this way typically display non-trivial dynamics: in particular, they are generally aperiodic with no obvious patterns. We discuss and illustrate this behavior with some examples.
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