Journal of Integer Sequences, Vol. 1 (1998), Article 98.1.8 |

Stephen J. Greenfield

and

Eugene R. Speer

Department of Mathematics,

Rutgers University, New Brunswick, NJ 08903-2390

Email addresses: duke@math.rutgers.edu,
greenfie@math.rutgers.edu,

speer@math.rutgers.edu

**Abstract:** The terms of A000278,
the sequence defined by h_{0} = 0, h_{1} = 1, and
h_{n+2} = h_{n+1} + h_{n}^{2}, count the
trees in certain recursively defined forests. We show that
for *n* large, h_{n} is approximately
A^{sqrt(2)n} for n even and h_{n} is
approximately B^{sqrt(2)n} for n odd, with A,B > 1
and A not equal to B, and we give estimates of A and B: A is 1.436
± .001 and B is 1.452 ± .001. The doubly exponential
growth of the sequence is not surprising (see, for example, [AS]) but
the dependence of the growth on the parity of the subscript is more
interesting. Numerical and analytical investigation of similar
sequences suggests a possible generalization of this result to a large
class of such recursions.

This paper was written in December 1997 with revisions written in February and May 1998, and was based primarily on work done in 1993. Published in Journal of Integer Sequences, October 30 1998.

Return to