A380980 Place 2n distinct positive integers on an n-gon, n on the vertices and n on the sides such that the sums of the three values on all sides are equal. a(n) is the minimal sum of all the integers used.

21, 38, 55, 81, 105, 140

Offset

3,1

Links

Table of n, a(n) for n=3..8.

Example

For n=7 the first 14 positive integers suffice. Their permutation is 1,11,7,8,4,9,6,10,3,14,2,12,5,13 and the sum on each side of the heptagon is 19.

Mathematica

mod[n_]:=Append[n, First[n]];
plus[n_]:=Table[mod[n][[i]]+mod[n][[i+1]], {i, 1, Length[mod[n]]-1}];
goodPermutations[n_]:=Select[Permutations[n], Length[plus[#]]==Length[Union[plus[#]]]&];
min[n_]:=Min[Max/@plus/@goodPermutations[Range[n]]];
bestPermutation[n_]:=Select[goodPermutations[Range[n]], Max[plus[#]]==min[n]&, 1];
plusBP[n_]:=plus/@bestPermutation[n]; max[n_]:=Max[Max/@plusBP[n]];
unit[n_]:=max[n]+n+1; sum[n_]:=n*unit[n]-Total@@plusBP[n]+n*(n+1)/2; sum/@Range[3, 8]

Crossrefs

Cf. A380853.

Sequence in context: A224701 A050782 A061906 * A139768 A307278 A176071

Adjacent sequences: A380975 A380976 A380977 * A380981 A380982 A380984

Keyword

nonn,more,new

Author

Ivan N. Ianakiev, Feb 10 2025

Status

approved