Fp -affine recurrent n-dimensional sequences over F q are p-automatic

• Authors: Mihai Prunescu
• Venue: European Journal of Combinatorics
• Year: 2013

A recurrent 2-dimensional sequence a(m,n) is given by fixing particular sequences a(m,0), a(0,n) as initial conditions and a rule of recurrence a(m,n)=f(a(m,n-1),a(m-1,n-1),a(m-1,n)) for m,n=1. We generalize this concept to an arbitrary number of dimensions and of predecessors. We give a criterion for a general n-dimensional recurrent sequence to be alternatively produced by an n-dimensional substitution - i.e. to be an automatic sequence. We show also that if the initial conditions are p-automatic and the rule of recurrence is an F"p-affine function, then the n-dimensional sequence is p-automatic. Consequently all such n-dimensional sequences can be also defined by n-dimensional substitution. Finally we show various positive examples, but also a 2-dimensional recurrent sequence which is not k-automatic for any k. As a byproduct we show that for polynomials f@?Q[X] with deg(f)=2 and f(N)@?N, the characteristic sequence of the set f(N) is not k-automatic for any k.