7 research outputs found
On formal inverse of the Prouhet-Thue-Morse sequence
Let be a prime number and consider a -automatic sequence and its generating function
. Moreover, let us
suppose that and and consider the formal power series
which is a compositional inverse of , i.e.,
. In this note we initiate the study of arithmetic
properties of the sequence of coefficients of the power series . We are
mainly interested in the case when , where
and is the
Prouhet-Thue-Morse sequence defined on the two letter alphabet . More
precisely, we study the sequence which is the
sequence of coefficients of the compositional inverse of the generating
function of the sequence . This sequence is clearly 2-automatic. We
describe the sequence characterizing solutions of the equation
. In particular, we prove that the sequence is 2-regular. We
also prove that an increasing sequence characterizing solutions of the equation
is not -regular for any . Moreover, we present a result
concerning some density properties of a sequence related to .Comment: 16 pages; revised version will appear in Discrete Mathematic