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Discrete Mathematics & Theoretical Computer Science ; vol. 22 no. 1 ; Automata, Logic and Semantics ; 1365-8050, A formal inverse of a given automatic sequence (the sequence of coefficients of the composition inverse of its associated formal power series) is also automatic. The comparison of properties of the original sequence and its formal inverse is an interesting problem. Such an analysis has been done before for the Thue{Morse sequence. In this paper, we describe arithmetic properties of formal inverses of the generalized Thue-Morse sequences and formal inverses of two modifications of the Rudin{Shapiro sequence. In each case, we give the recurrence relations and the automaton, then we analyze the lengths of strings of consecutive identical letters as well as the frequencies of letters. We also compare the obtained results with the original sequences., Comment: 20 pages

In a recent preprint on ArXiv, Bacher introduced a twisted version of the Stern sequence. His paper contains in particular three conjectures relating the generating series for the Stern sequence and for the twisted Stern sequence. Soon afterwards Coons published two papers in {\it Integers}: first he proved these conjectures, second he used his result to obtain a correlation-type identity for the Stern sequence. We recall here a simple result of Reznick and we state a similar result for the twisted Stern sequence. We deduce an easy proof of Coons' identity, and a simple proof of Bacher's conjectures. Furthermore we prove identities similar to Coons' for variations on the Stern sequence that include Bacher's sequence.