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A070085 a(n) = A070080(n)^2 + A070081(n)^2 - A070082(n)^2.

%I A070085 #14 Feb 16 2025 08:32:46
%S A070085 1,1,4,1,-1,4,1,-3,9,4,2,1,-5,-7,9,4,0,16,1,-7,-11,9,7,4,-2,-4,16,1,
%T A070085 -9,-15,9,-17,5,25,4,-4,-8,16,14,1,-11,-19,9,-23,3,1,25,4,-6,-12,16,
%U A070085 -14,12,36,1,-13,-23,9,-29,1,-31,-3,25,23,4,-8
%N A070085 a(n) = A070080(n)^2 + A070081(n)^2 - A070082(n)^2.
%C A070085 The integer triangle [A070080(n)<=A070081(n)<=A070082(n)] is acute iff a(n)>0, right iff a(n)=0 and obtuse iff a(0)<0.
%H A070085 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AcuteTriangle.html">Acute Triangle</a>.
%H A070085 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RightTriangle.html">Right Triangle</a>.
%H A070085 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ObtuseTriangle.html">Obtuse Triangle</a>.
%H A070085 R. Zumkeller, <a href="/A070080/a070080.txt">Integer-sided triangles</a>
%t A070085 maxPer = m = 22;
%t A070085 sides[per_] := Select[Reverse /@ IntegerPartitions[per, {3}, Range[Ceiling[ per/2]]], #[[1]] < per/2 && #[[2]] < per/2 && #[[3]] < per/2&];
%t A070085 triangles = DeleteCases[Table[sides[per], {per, 3, m}], {}] // Flatten[#, 1]& // SortBy[Total[#] m^3 + #[[1]] m^2 + #[[2]] m + #[[1]]&];
%t A070085 #[[1]]^2 + #[[2]]^2 - #[[3]]^2& /@ triangles (* _Jean-François Alcover_, Jul 31 2018 *)
%Y A070085 Cf. A070093, A070101, A024155, A070118, A070127, A070136.
%K A070085 sign
%O A070085 1,3
%A A070085 _Reinhard Zumkeller_, May 05 2002