A070149 Areas of integer Heronian triangles [A070080(A070142(n)), A070081(A070142(n)), A070082(A070142(n))].

6, 12, 12, 24, 30, 24, 48, 36, 54, 48, 60, 60, 42, 84, 66, 84, 96, 108, 60, 120, 36, 90, 126, 108, 84, 60, 120, 150, 72, 96, 168, 120, 192, 132, 204, 210, 210, 84, 144, 216, 192, 240, 114, 156, 180, 120, 240, 300, 168, 210, 168

Offset: 1

Author

Reinhard Zumkeller, May 05 2002

Keywords

• nonn

Examples

			A070142(2)=39: [A070080(39), A070081(39), A070082(39)] = [5,5,6], area^2 = s*(s-5)*(s-5)*(s-6) with s=A070083(39)/2=(5+5+6)/2=8, area^2=8*3*3*2=16*9 is an integer square, therefore a(2)=A070086(39)=area=4*3=12.
		

Links

• Jean-François Alcover, Table of n, a(n) for n = 1..1265
• Eric Weisstein's World of Mathematics, Heronian Triangle.
• Reinhard Zumkeller, Integer-sided triangles

Crossrefs

Cf. A070142, A070210.

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]] &];
area[{a_, b_, c_}] := With[{p = (a+b+c)/2}, Sqrt[p(p-a)(p-b)(p-c)]];
Select[area /@ triangles, IntegerQ] (* Jean-François Alcover, Oct 12 2021 *)

Formula

a(n) = A070086(A070142(n)).