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The following was conjectured by D. Reutter in problem 644A664A, Elemente der mathematik 27 and proved by H. Harborth in Solution to problem 644A664A, Elemente der mathematik 29, 14-15

The maximum number of times the minimum distance can occur among $n$ points in the plane is $\lfloor 3n-\sqrt{12n-3}\rfloor$.

This is achieved by a hexagonal piece in the triangular lattice, i.e. from points forming a hexagonal spiral. In particular, this gives a formula for your first sequence.
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The following was conjectured by D. Reutter in problem 644A, Elemenete Elemente der mathematik 27 and proved by H. Harborth in Solution to problem 644A, Elemente der mathematik 29, 14-15

The maximum number of times the minimum distance can occur among $n$ points in the plane is $\lfloor 3n-\sqrt{12n-3}\rfloor$.

This is achieved by a hexagonal piece in the triangular lattice, i.e. from points forming a hexagonal spiral. In particular, this gives a formula for your first sequence.
show/hide this revision's text 1

The following was conjectured by D. Reutter in problem 644A, Elemenete der mathematik 27 and proved by H. Harborth in Solution to problem 644A, Elemente der mathematik 29, 14-15

The maximum number of times the minimum distance can occur among $n$ points in the plane is $\lfloor 3n-\sqrt{12n-3}\rfloor$.

This is achieved by a hexagonal piece in the triangular lattice, i.e. from points forming a hexagonal spiral. In particular, this gives a formula for your first sequence.