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).
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.