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A070080 Smallest side of integer triangles [a(n) <= A070081(n) <= A070082(n)], sorted by perimeter, lexicographically ordered.

%I A070080 #31 Oct 04 2021 07:11:32
%S A070080 1,1,2,1,2,2,1,2,3,2,3,1,2,3,3,2,3,4,1,2,3,3,4,2,3,4,4,1,2,3,3,4,4,5,
%T A070080 2,3,4,4,5,1,2,3,3,4,4,5,5,2,3,4,4,5,5,6,1,2,3,3,4,4,5,5,5,6,2,3,4,4,
%U A070080 5,5,6,6,1,2,3,3,4,4,5,5,5,6,6,7,2,3,4,4,5,5
%N A070080 Smallest side of integer triangles [a(n) <= A070081(n) <= A070082(n)], sorted by perimeter, lexicographically ordered.
%H A070080 G. C. Greubel, <a href="/A070080/b070080.txt">Table of n, a(n) for the first 55 rows</a>
%H A070080 Reinhard Zumkeller, <a href="/A070080/a070080.txt">Integer-sided triangles</a>
%F A070080 a(n) = A070083(n) - A070082(n) - A070081(n).
%t A070080 m = 55 (* max perimeter *);
%t A070080 sides[per_] := Select[Reverse /@ IntegerPartitions[per, {3}, Range[ Ceiling[per/2]]], #[[1]] < per/2 && #[[2]] < per/2 && #[[3]] < per/2&];
%t A070080 triangles = DeleteCases[Table[sides[per], {per, 3, m}], {}] // Flatten[#, 1]& // SortBy[Total[#] m^3 + #[[1]] m^2 + #[[2]] m + #[[1]]&];
%t A070080 triangles[[All, 1]] (* _Jean-François Alcover_, Jun 12 2012, updated Jul 09 2017 *)
%Y A070080 Cf. A046128, A055594, A069597.
%Y A070080 Cf. A070081, A070082, A070083, A070084, A070085, A070086.
%Y A070080 Cf. A316841, A316843, A316844, A316845 (sides (i,j,k) with j + k > i >= j >= k >= 1).
%Y A070080 Cf. A331244, A331245, A331246 (similar, but triangles sorted by radius of enclosing circle), A331251, A331252, A331253 (triangles sorted by area), A331254, A331255, A331256 (triangles sorted by radius of circumcircle).
%K A070080 nonn
%O A070080 1,3
%A A070080 _Reinhard Zumkeller_, May 05 2002