Self-generating sets, integers with missing blocks, and substitutions

摘要

We give a new construction of the Kimberling sequence defined by: (a) 1 belongs to S; (b) if the positive integer x belongs to S, then 2x and 4x-1 belong to S; and (c) nothing else belongs to S, hence, S = 1 2 3 4 6 7 8 11 12 14 15 16 ... which is sequence A052499 in the Sloane's On-line Encyclopedia of Integer Sequences, by proving that this sequence is equal to sequence 1+ A003754, the sequence of integers whose binary expansion does not contain the block of digits 00. We give a general framework for this sequence and similar sequences, in relation to automatic or morphic sequences and to non-standard numeration systems such as the lazy Fibonacci expansion.