A070136 Numbers m such that [A070080(m), A070081(m), A070082(m)] is a right integer triangle.

17, 116, 212, 370, 493, 850, 1297, 1599, 1629, 2574, 2778, 3751, 4298, 4370, 5251, 5286, 6476, 9169, 10066, 12398, 12441, 12520, 14414, 16365, 16602, 19831, 21231, 21486, 24060, 26125, 27245, 29230, 33625, 33658

Offset: 1

Author

Reinhard Zumkeller, May 05 2002

Keywords

• nonn

Comments

Right integer triangles have integer areas: see A070142.

Examples

			116 is a term: [A070080(116), A070081(116), A070082(116)]=[6,8,10], A070085(116)=6^2+8^2-10^2=36+64-100=0.
212 is a term: [A070080(212), A070081(212), A070082(212)]=[5,12,13], A070085(212)=5^2+12^2-13^2=25+144-169=0.
		

Links

• Jean-François Alcover, Table of n, a(n) for n = 1..137
• Eric Weisstein's World of Mathematics, Heronian Triangle.
• Eric Weisstein's World of Mathematics, Right Triangle.
• Reinhard Zumkeller, Integer-sided triangles

Crossrefs

Cf. A024155, A070137, A070142, A070085.

Programs

Mathematica

m = 500 (* max perimeter *);
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]] &];
Position[triangles, {a_, b_, c_} /; a^2 + b^2 == c^2] // Flatten (* Jean-François Alcover, Oct 12 2021 *)