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Summary: On Integer Sequences Whose First Iterates are Linear
Jean-Paul Allouche
CNRS, LRI
B^atiment 490
F-91405 Orsay Cedex
France
allouche@lri.fr
Narad Rampersad and Je rey Shallit
School of Computer Science
University of Waterloo
Waterloo, Ontario N2L 3G1
Canada
nrampersad@math.uwaterloo.ca
shallit@graceland.uwaterloo.ca
Abstract
In this paper we discuss the functional equation a(a(n)) = dn, where (a(n)) n0
is a monotone 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
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