Reverend Back's Abbey Floor

Consider the sequence defined by  and , where  denotes the reverse of a sequence . The first few terms are then 01, 010110, 010110010110011010, .... All words  are cubefree (Allouche and Shallit 2003, p. 28, Ex. 1.49). Iterating gives the sequence 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, ... (OEIS A118006)

Plotting  (mod 2), where  denotes the th digit of the infinitely iterated sequence, gives the beautiful pattern shown above, known as Reverend Back's abbey floor (Wegner 1982; Siromoney and Subramanian 1983; Allouche and Shallit 2003, pp. 410-411). Note that this plot is identical to the recurrence plot  (mod 2).

REFERENCES:

Allouche, J.-P. and Shallit, J. Automatic Sequences: Theory, Applications, Generalizations. Cambridge, England: Cambridge University Press, 2003.

Siromoney, R. and Subramanian, K. G. "Generative Grammar for the Cube-Free Abbey Floor." Bull. Eur. Assoc. Theor. Comput. Sci., No. 20, 160-162, Jun. 1983.

Sloane, N. J. A. Sequence A118006 in "The On-Line Encyclopedia of Integer Sequences."

Wegner, L. "Problem P12: Is  Cube-Free?" Bull. Eur. Assoc. Theor. Comput. Sci., No. 18, 120, Oct. 1982.