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

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, -9, -15, 9, -17, 5, 25, 4, -4, -8, 16, 14, 1, -11, -19, 9, -23, 3, 1, 25, 4, -6, -12, 16, -14, 12, 36, 1, -13, -23, 9, -29, 1, -31, -3, 25, 23, 4, -8

Offset: 1

Author

Reinhard Zumkeller, May 05 2002

Keywords

• sign

Comments

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.

Links

• Eric Weisstein's World of Mathematics, Acute Triangle.
• Eric Weisstein's World of Mathematics, Right Triangle.
• Eric Weisstein's World of Mathematics, Obtuse Triangle.
• R. Zumkeller, Integer-sided triangles

Crossrefs

Cf. A070093, A070101, A024155, A070118, A070127, A070136.

Programs

Mathematica

maxPer = m = 22;
sides[per_] := Select[Reverse /@ IntegerPartitions[per, {3}, Range[Ceiling[ per/2]]], #[[1]] < per/2 && #[[2]] < per/2 && #[[3]] < per/2&];
triangles = DeleteCases[Table[sides[per], {per, 3, m}], {}] // Flatten[#, 1]& // SortBy[Total[#] m^3 + #[[1]] m^2 + #[[2]] m + #[[1]]&];
#[[1]]^2 + #[[2]]^2 - #[[3]]^2& /@ triangles (* Jean-François Alcover, Jul 31 2018 *)