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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 A380980 #15 Feb 13 2025 08:26:28
%S A380980 21,38,55,81,105,140
%N 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.
%e A380980 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 A380980 mod[n_]:=Append[n,First[n]];
%t A380980 plus[n_]:=Table[mod[n][[i]]+mod[n][[i+1]],{i,1,Length[mod[n]]-1}];
%t A380980 goodPermutations[n_]:=Select[Permutations[n],Length[plus[#]]==Length[Union[plus[#]]]&];
%t A380980 min[n_]:=Min[Max/@plus/@goodPermutations[Range[n]]];
%t A380980 bestPermutation[n_]:=Select[goodPermutations[Range[n]],Max[plus[#]]==min[n]&,1];
%t A380980 plusBP[n_]:=plus/@bestPermutation[n]; max[n_]:=Max[Max/@plusBP[n]];
%t A380980 unit[n_]:=max[n]+n+1; sum[n_]:=n*unit[n]-Total@@plusBP[n]+n*(n+1)/2; sum/@Range[3,8]
%Y A380980 Cf. A380853.
%K A380980 nonn,more
%O A380980 3,1
%A A380980 _Ivan N. Ianakiev_, Feb 10 2025