Conway studied the following recurrence relation (OEIS), purportedly studied originally by Hofstadter (of G.E.B. fame):
$$a(k)=a(a(k-1))+a(k-(a(k-1)))$$
with initial conditions $a(1)=a(2)=1$.
(image from MathWorld)
Conway was able to show that
$$\lim_{k\to\infty}\frac{a(k)}{k}=\frac12$$
He offered a \$10,000 prize to anyone who could discover a value of $k$ such that
$$\left|\frac{a(j)}{j}-\frac12\right|<\frac1{20},\quad j > k$$
Collin Mallows from Bell Labs found $k=3173375556$, 34 days after Conway's initial talk on the sequence, and the prize was awarded by Conway after "adjusting" it to the "intended" value of \$1,000.
