A380980 - OEIS

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.

%I #15 Feb 13 2025 08:26:28

%S 21,38,55,81,105,140

%N 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.

%e 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.

%t mod[n_]:=Append[n,First[n]];

%t plus[n_]:=Table[mod[n][[i]]+mod[n][[i+1]],{i,1,Length[mod[n]]-1}];

%t goodPermutations[n_]:=Select[Permutations[n],Length[plus[#]]==Length[Union[plus[#]]]&];

%t min[n_]:=Min[Max/@plus/@goodPermutations[Range[n]]];

%t bestPermutation[n_]:=Select[goodPermutations[Range[n]],Max[plus[#]]==min[n]&,1];

%t plusBP[n_]:=plus/@bestPermutation[n]; max[n_]:=Max[Max/@plusBP[n]];

%t unit[n_]:=max[n]+n+1; sum[n_]:=n*unit[n]-Total@@plusBP[n]+n*(n+1)/2; sum/@Range[3,8]

%Y Cf. A380853.

%K nonn,more,new

%O 3,1

%A _Ivan N. Ianakiev_, Feb 10 2025