... is often called the Thue-Morse sequence and has appeared in variousfields of mathematics. See, for example, the paper [AS1], which contains a review of themain properties of this sequence and ... =k∈N1k!Gk.An explicit formula. It is interesting to give an alternate (longer) proof of Lemma 7,which has the advantage of providing the explicit formulas of H and G. There are twosteps. ... Topics in matrix analysis, Cambridge university press,1991.[Ma] B. B. Mandelbrot, The fractal geometry of nature, W. H. Freeman and Co., 1977.[Sh] J. Shallit, A generalization of automatic sequences,...