Silverman's Sequence

Let $f(1)=1$, and let $f(n)$ be the number of occurrences of $n$ in a nondecreasing sequence of Integers. Then the first few values of $f(n)$ are 1, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, ... (Sloane's A001462). The asymptotic value of the $n$th term is $\phi^{2-\phi}n^{\phi-1}$, where $\phi$ is the Golden Ratio.

References

Guy, R. K. ``Silverman's Sequences.'' §E25 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 225-226, 1994.

Sloane, N. J. A. Sequence A001462/M0257 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.