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Reverend Back's Abbey Floor


Consider the sequence defined by w_1=01 and w_(n+1)=w_nw_nw_n^R, where l^R denotes the reverse of a sequence l. The first few terms are then 01, 010110, 010110010110011010, .... All words w_n 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)

ReverendBacksAbbeyFloor

Plotting w_infty(x)+w_infty(y) (mod 2), where w_infty(n) denotes the nth 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 w_infty(x)-w_infty(y) (mod 2).


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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 www^R Cube-Free?" Bull. Eur. Assoc. Theor. Comput. Sci., No. 18, 120, Oct. 1982.

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Reverend Back's Abbey Floor

Cite this as:

Weisstein, Eric W. "Reverend Back's Abbey Floor." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ReverendBacksAbbeyFloor.html

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