Journal of Integer Sequences, Vol. 21 (2018), Article 18.9.1

## The Formal Inverse of the Period-Doubling Sequence

### Narad Rampersad Department of Mathematics and Statistics University of Winnipeg 515 Portage Avenue Winnipeg, Manitoba R3B 2E9 Canada Manon Stipulanti Département de Mathématique University of Liège Allée de la Découverte 12 (B37) 4000 Liège Belgium

Abstract:

If p is a prime number, consider a p-automatic sequence (un)n≥0, 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 ∈ Fp[[X]] which is the compositional inverse of U, i.e., U(V(X)) = X = V(U(X)). The problem investigated in this paper is to study the properties of the sequence (vn)n≥0. The work was first initiated for the Thueâ€“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.

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(Concerned with sequences A079523 A086747 A096268 A121539 A317542 A317543 A317544.)

Received September 10 2018; revised versions received December 5 2018. Published in Journal of Integer Sequences, December 5 2018.