The Formal Inverse of the Period-Doubling Sequence
Department of Mathematics and Statistics
University of Winnipeg
515 Portage Avenue
Winnipeg, Manitoba R3B 2E9
Département de Mathématique
University of Liège
Allée de la Découverte 12
(B37) 4000 Liège
If p is a prime number,
consider a p-automatic sequence
and let U(X) = Σn≥0 unXn ∈ Fp[[X]]
be its generating function.
Assume that there exists a formal power series
V(X) = Σn≥0 vnXn
which is the compositional inverse of
U(V(X)) = X = V(U(X)).
The problem investigated in this paper is to study the properties of the
The work was first initiated for the Thue–Morse
sequence, and more recently the case of other sequences (variations of
the Baum-Sweet sequence, variations of the Rudin-Shapiro sequence
and generalized Thue-Morse sequences) has been treated. In this paper,
we deal with the case of the period-doubling sequence. We first show that
the sequence of indices at which the period-doubling sequence takes the
value 0 (resp., 1) is not k-regular for any k ≥ 2.
Secondly, we give
recurrence relations for its formal inverse, then we show that it is
2-automatic, and we also provide an automaton that generates it. Thirdly,
we study the sequence of indices at which this formal inverse takes
the value 1, and we show that it is not
k-regular for any k ≥ 2 by
connecting it to the characteristic sequence of Fibonacci numbers. We
leave as an open problem the case of the sequence of indices at which
this formal inverse takes the value 0.
Full version: pdf,
(Concerned with sequences
Received September 10 2018; revised versions received December 5 2018.
Published in Journal of Integer Sequences, December 5 2018.
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