Journal of Integer Sequences, Vol. 11 (2008), Article 08.2.1 |

Department of Mathematics

University of Toronto

Toronto, Ontario M5S 2E4

Canada

**Abstract:**

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

(Concerned with sequences A008619 and A109964 .)

Received January 4 2008;
revised version received May 22 2008.
Published in *Journal of Integer Sequences*, May 23 2008.

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