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Areas of integer-sided triangles whose area equals 3 times their perimeter.
7

%I #11 Aug 06 2017 14:27:40

%S 192,204,210,216,240,252,264,270,324,330,336,378,384,408,420,456,462,

%T 480,504,522,540,546,624,690,714,780,792,840,876,966,990,1176,1248,

%U 1320,1380,1806,2394,2460,3120,4446,8436

%N Areas of integer-sided triangles whose area equals 3 times their perimeter.

%C There are no further terms.

%C For a(3)=210, there are 2 solutions (20,21,29),(17,25,28);

%C For a(11)=336, there are 2 solutions (14,48,50),(24,35,53);

%C For a(16)=456, a(22)=546, there are 2 solutions respectively too.

%e The areas 192,204,210,216,240, ... pertain respectively to triangles with sides (20,20,24), (17,25,26), (20,21,29), (18,24,30), (16,30,34), ..., equal 3 times their perimeter 64,68,70,72,80, ...

%t f[a_, b_, c_] := Block[{P = Total[{a, b, c}]/2}, Sqrt[P (P - a) (P - b) (P - c)]]; Sort@ Map[f @@ # &, Select[Union@ Map[Sort, Tuples[Range@ 150, {3}]], f @@ # == 3 Total@# &] ] (* _Michael De Vlieger_, Jul 03 2017 *)

%Y Cf. A098030, A289218.

%Y A row of the triangle in A290451.

%K nonn,fini,full

%O 1,1

%A _Zhining Yang_, Jun 28 2017