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%I #15 Oct 12 2021 21:57:41
%S 6,12,12,24,30,24,48,36,54,48,60,60,42,84,66,84,96,108,60,120,36,90,
%T 126,108,84,60,120,150,72,96,168,120,192,132,204,210,210,84,144,216,
%U 192,240,114,156,180,120,240,300,168,210,168
%N Areas of integer Heronian triangles [A070080(A070142(n)), A070081(A070142(n)), A070082(A070142(n))].
%H Jean-François Alcover, <a href="/A070149/b070149.txt">Table of n, a(n) for n = 1..1265</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HeronianTriangle.html">Heronian Triangle</a>.
%H Reinhard Zumkeller, <a href="/A070080/a070080.txt">Integer-sided triangles</a>
%F a(n) = A070086(A070142(n)).
%e 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.
%t m = 500 (* max perimeter *);
%t sides[per_] := Select[Reverse /@ IntegerPartitions[per, {3}, Range[ Ceiling[per/2]]], #[[1]] < per/2 && #[[2]] < per/2 && #[[3]] < per/2 &];
%t triangles = DeleteCases[Table[sides[per], {per, 3, m}], {}] // Flatten[#, 1]& // SortBy[Total[#] m^3 + #[[1]] m^2 + #[[2]] m + #[[1]] &];
%t area[{a_, b_, c_}] := With[{p = (a+b+c)/2}, Sqrt[p(p-a)(p-b)(p-c)]];
%t Select[area /@ triangles, IntegerQ] (* _Jean-François Alcover_, Oct 12 2021 *)
%Y Cf. A070142, A070210.
%K nonn
%O 1,1
%A _Reinhard Zumkeller_, May 05 2002