Formula omitted]-affine recurrent [formula omitted]-dimensional sequences over [formula omitted] are [formula omitted]-automatic

Abstract

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 Fp-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.