  Previously published packings of equal disks in an equilateral triangle have
dealt with up to 21 disks. We use a new discrete-event simulation algorithm to
produce packings for up to 34 disks. For each n in the range 22 =< n =< 34 we
present what we believe to be the densest possible packing of n equal disks in
an equilateral triangle. For these n we also list the second, often the third
and sometimes the fourth best packings among those that we found. In each case,
the structure of the packing implies that the minimum distance d(n) between
disk centers is the root of polynomial P_n with integer coefficients. In most
cases we do not explicitly compute P_n but in all cases we do compute and
report d(n) to 15 significant decimal digits. Disk packings in equilateral
triangles differ from those in squares or circles in that for triangles there
are an infinite number of values of n for which the exact value of d(n) is
known, namely, when n is of the form Delta(k) := k(k+1)/2. It has also been
conjectured that d(n-1) = d(n) in this case. Based on our computations, we
present conjectured optimal packings for seven other infinite classes of n,
namely n = Delta(2k)+1, Delta(2k+1)+1, Delta(k+2)-2, Delta(2k+3)-3,
Delta(3k+1)+2, 4*Delta(k) and 2*Delta(k+1)+2*Delta(k)-1. We also report the
best packings we found for other values of n in these forms which are larger
than 34, namely, n=37, 40, 42, 43, 46, 49, 56, 57, 60, 63, 67, 71, 79, 84, 92,
93, 106, 112, 121, and 254, and also for n=58, 95, 108, 175, 255, 256, 258, and
260. We say that an infinite class of packings of n disks, n=n(1),
n(2),...n(k),..., is tight, if 1/d(n(k)+1) - 1/d(n(k)) is bounded away from
zero as k goes to infinity. We conjecture that some of our infinite classes are
tight, others are not tight, and that there are infinitely many tight classes.
