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3 edited body

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.

2 Correct typo: stray e in first Elemente

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.

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.