Sequence from Stan Wagon's Macalester POW #1060

[Joshua Zucker]

The problem:
Given k coins, s of which are heavy,
and a scale which, given a stack of coins on it, either reads "normal"
(if all are normal) or "heavy" (if at least one is heavy),
what is the fewest weighings to identify the which coins are the heavy ones?

(Alternative interpretation: Given k blood samples, s of which are
diseased, and a test which can take arbitrarily small bits of sample
and tell whether disease is present or not, identify which s samples
are the diseased ones.)

If s is 1, then binary search is the best, so the number of weighings
it takes is ceiling (log base 2 of k).

That's A029837.

If s is 2, then apparently even for k as small as 22, nobody knows for
sure whether the answer is 8 or 9.

Does anyone have an elegant solution to the s = 2 problem?  Or a
sequence that gives the first known terms?  Or should I submit one
with my best efforts?

Stan Wagon's problem is k = 1,000,000 and s = 100, and there is the
obvious entropy lower bound, and some algorithms given that get fairly
close to that lower bound, but still the best we can say is that the
answer is "around 1500" from the information I've seen.

--Joshua Zucker