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A232461 Integer areas of integer-sided triangles where two sides are of square length.

%I A232461 #17 Feb 19 2019 18:18:41
%S A232461 120,168,300,360,1920,2016,2688,4680,4800,5760,9720,10140,13608,14280,
%T A232461 18720,19080,23256,24300,29160,30720,32760,34440,34680,38640,42120,
%U A232461 43008,57720,74880,75000,76800,92160,94080,105000,128700,162240,177072,187500,217728
%N A232461 Integer areas of integer-sided triangles where two sides are of square length.
%C A232461 Subset of A188158.
%C A232461 The areas of the triangles (a,b,c) are given by Heron's formula, A = sqrt(s(s-a)(s-b)(s-c)), where its side lengths are a, b, c and semiperimeter s = (a+b+c)/2.
%C A232461 The areas A of the primitive triangles of sides (a,b,c) are 120, 168, 300, 360, 4680, ...
%C A232461 The areas of the nonprimitive triangle of sides (a*p^2, b*p^2, c*p^2) are in the sequence with the value A*p^4.
%C A232461 It is possible to find integer-sided triangles having two square sides, for example:
%C A232461 a(2) = 168 with sides (25,25,48) and (14,25,25);
%C A232461 a(3) = 300 with sides (25,25,30) and (25,25,40);
%C A232461 a(14) = 14280 with sides (169,169,238), (169,169,240), (100,289,291).
%C A232461 The following table gives the first values (A, a, b, c):
%C A232461 +-------+-----+-----+-----+
%C A232461 |   A   |  a  |  b  |  c  |
%C A232461 +-------+-----+-----+-----+
%C A232461 |   120 |  16 |  25 |  39 |
%C A232461 |   168 |  14 |  25 |  25 |
%C A232461 |   300 |  25 |  25 |  40 |
%C A232461 |   360 |  25 |  29 |  36 |
%C A232461 |  1920 |  64 | 100 | 156 |
%C A232461 |  2016 |  64 | 225 | 287 |
%C A232461 |  2688 | 100 | 100 | 192 |
%C A232461 |  4680 |  74 | 169 | 225 |
%C A232461 |  4800 | 100 | 100 | 160 |
%C A232461 |  5760 | 100 | 116 | 144 |
%C A232461 |  9720 | 144 | 225 | 351 |
%C A232461 | 10140 | 169 | 169 | 312 |
%C A232461 | 13608 | 225 | 225 | 432 |
%C A232461 +-------+-----+-----+-----+
%H A232461 J. Peng, Y. Zhang, <a href="https://doi.org/10.1007/s10474-018-00907-0">Heron triangles with figurate number sides</a>, Acta Mathematica Hungarica (2019) 1-11.
%e A232461 120 is in the sequence because the triangle (4^2, 5^2, 39) has semiperimeter s = (16+25+39)/2 = 40, and A = sqrt(40*(40-16)*(40-25)*(40-39)) = 120.
%t A232461 nn=1000;lst={};Do[s=(a+b+c)/2;If[IntegerQ[s],area2=s (s-a) (s-b) (s-c);If[0<area2&&((IntegerQ[Sqrt[a]]&&IntegerQ[Sqrt[b]])||(IntegerQ[Sqrt[a]]&&IntegerQ[Sqrt[c]])||(IntegerQ[Sqrt[b]]&&IntegerQ[Sqrt[c]]))&&IntegerQ[Sqrt[area2]],AppendTo[lst,Sqrt[area2]]]],{a,nn},{b,a},{c,b}];Union[lst]
%t A232461 sqr[n_]:=n^2; area[a_, b_, c_] := Module[{s = (a + b + c)/2, a2}, a2 = s (s - a) (s - b) (s - c); If[a2 < 0, 0, Sqrt[a2]]]; goodQ[a_, b_, c_] := Module[{ar = area[a, b, c]}, ar > 0 && IntegerQ[ar]]; nn = 80; t = {}; ps = sqr[Range[2, nn]]; mx = 3*ps[[-1]]; Do[If[p <= q && goodQ[p, q, e], aa = area[p, q, e]; If[aa <= mx, AppendTo[t, aa]]], {p, ps}, {q, ps}, {e, q - p + 2, p + q - 2, 2}]; t = Union[t] (* program from _T. D. Noe_ adapted for this sequence - see A229746 *)
%Y A232461 Cf. A000290, A188158.
%K A232461 nonn
%O A232461 1,1
%A A232461 _Michel Lagneau_, Nov 24 2013