We give a quick overview of some basic classes of stream differential equations (SDEs), show that the collection of automatic sequences carries a final coalgebra structure, and use this to present new types of SDEs for them, using even, odd and zip.
Related references:
C. Kupke and J. Rutten. On the final coalgebra of automatic sequences. In Robert L. Constable and Alexandra Silva (eds.), Logic and Program Semantics. Essays dedicated to Dexter Kozen on the occasion of his 60th Birthday. LNCS 7230, pp. 149-164, Springer, 2012.
M. Bonsangue, J. Rutten and J. Winter. Defining context-free power series coalgebraically. In: D. Pattinson and L. Schroeder (Eds.), Proceedings of CMCS 2012, LNCS 7399, 2012, pp. 20--39.