A two-valued recurrent double sequence that is not automatic

A recurrent 2-dimensional (double) sequence t(m,n) is given by fixing particular sequences t(m, 0), t(0, n) as initial conditions and a rule of recurrence t(m,n) = t(a(m,n − 1), t(m − 1, n − 1), t(m − 1, n)) for m,n ≥ 1. We display such a sequence with values in the set {0, 1} and show that all rows are periodic and that the minimal period of the n-th row has length 2. We conclude that this 2-dimensional sequence is not k, l-automatic for any k, l ≥ 2. Other two-valued recurrent double sequences are shortly discussed. Minimal examples of non-trivial 2-automatic sequences are displayed. One case remains open.