Mephisto Waltz Sequence

The Mephisto waltz sequence is defined by beginning with 0 and then iterating the maps  and . This gives 0, 001, 001001110, 001001110001001110110110001, ... (OEIS A064990). These words are fourth power-free (Allouche and Shallit 2003, p. 25).

The numbers of 0s and 1s in step , 1, ... are given by 1, 2, 5, 14, 41, 122, ... (OEIS A007051) and 0, 1, 4, 13, 40, 121, ... (OEIS A003462), respectively, which are given in closed form by  and , respectively.

A recurrence plot of the Mephisto waltz sequence is illustrated above.

REFERENCES:

Allouche, J.-P. and Shallit, J. Exercise 1.9.16 in Automatic Sequences: Theory, Applications, Generalizations. Cambridge, England: Cambridge University Press, p. 25, 2003.

Jacobs, K. Invitation to Mathematics. Princeton, NJ: Princeton University Press, pp. 105-106 and 215, 1992.

Sloane, N. J. A. Sequences A007051/M1458, A003462/M3463, and A064990 in "The On-Line Encyclopedia of Integer Sequences."