A Combinatorial Interpretation for Certain Relatives of the Conolly Sequence

For any integer s ≥ 0, we derive a combinatorial interpretation for the family of sequences generated by the recursion (parameterized by s) hs(n) = hs(n − s − hs(n − 1)) + hs(n − 2− s − hs(n − 3)), n > s + 3, with the initial conditions hs(1) = hs(2) = · · · = hs(s+2) = 1 and hs(s+3) = 2. We show how these sequences count the number of leaves of a certain infinite tree structure. Using this interpretation we prove that hs sequences are “slowly growing”, that is, hs sequences are monotone nondecreasing, with successive terms increasing by 0 or 1, so each sequence hits every positive integer. Further, for fixed s the sequence hs(n) hits every positive integer twice except for powers of 2, all of which are hit s+2 times. Our combinatorial interpretation provides a simple approach for deriving the ordinary generating functions for these sequences.