Summary.
In this paper we discuss the functional equation a(a(n)) = dn, where (a(n))n≥0 is an increasing sequence of non-negative integers. Mallows observed this equation has a unique solution for d = 2, and Propp observed the same thing for d = 3. We show that the equation has uncountably many solutions for d ≥ 4. Further, we give a complete description for the lexicographically least such sequence, showing that the first difference sequence can be generated by a finite automaton.
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Manuscript received: March 19, 2003 and, in final form, June 25, 2004.
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Allouche, JP., Rampersad, N. & Shallit, J. On integer sequences whose first iterates are linear. Aequ. math. 69, 114–127 (2005). https://doi.org/10.1007/s00010-004-2750-x
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DOI: https://doi.org/10.1007/s00010-004-2750-x