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A256695 Integer areas A of integer-sided triangles (a, b, c) such that the area of the triangle (a+b, a+c, b+c) is also an integer.

%I A256695 #20 Aug 08 2025 01:54:29
%S A256695 12,48,108,192,300,432,588,768,972,1008,1200,1452,1728,2028,2352,2448,
%T A256695 2520,2700,2772,3060,3072,3468,3888,4032,4332,4800,5292,5808,6348,
%U A256695 6912,7500,8112,8748,9072,9408,9792,10080,10092,10800,11088,11532,11628,12240,12288
%N A256695 Integer areas A of integer-sided triangles (a, b, c) such that the area of the triangle (a+b, a+c, b+c) is also an integer.
%C A256695 The areas of the primitive triangles are 12, 2520, 2772, 3060, 4032, 5808, 9072, 11088, 11628, 17136, 24948, 41580, ...
%C A256695 The area A of a triangle whose sides have lengths a, b, and c is given by Heron's formula: A = sqrt(s*(s-a)*(s-b)*(s-c)), where s = (a+b+c)/2.
%C A256695 The sequence A135453 (numbers of the form 12*n^2) is included in the sequence because a(1) = 12 is a primitive triangle of the subsequence k^2*a(1), k=1,2,3,...
%C A256695 The following table gives the first values (A, A', a, b, c) where A is the integer area of the triangle (a, b, c), A' is the integer area of the triangle (a+b, a+c, b+c).
%C A256695 +-------+--------+------+------+------+
%C A256695 |    A  |    A'  |   a  |   b  |   c  |
%C A256695 +-------+--------+------+------+------+
%C A256695 |   12  |    60  |   5  |   5  |   8  |
%C A256695 |   48  |   240  |  10  |  10  |  16  |
%C A256695 |  108  |   540  |  15  |  15  |  24  |
%C A256695 |  192  |   960  |  20  |  20  |  32  |
%C A256695 |  300  |  1500  |  25  |  25  |  40  |
%C A256695 |  432  |  2160  |  30  |  30  |  48  |
%C A256695 |  588  |  2940  |  35  |  35  |  56  |
%C A256695 |  768  |  3840  |  40  |  40  |  64  |
%C A256695 |  972  |  4860  |  45  |  45  |  72  |
%C A256695 +-------+--------+------+------+------+
%C A256695 We find a majority of isosceles triangles, but there is a subsequence of non-isosceles triangles with areas 2520, 3060, 10080, 11088, ...
%C A256695 +--------+----------+-------+-------+-------+
%C A256695 |    A   |     A'   |   a   |   b   |   c   |
%C A256695 +--------+----------+-------+-------+-------+
%C A256695 |   2520 |   18270  |   29  |  174  |  175  |
%C A256695 |   3060 |   33150  |   39  |  221  |  250  |
%C A256695 |  10080 |   73080  |   58  |  348  |  350  |
%C A256695 |  11088 |   64350  |  150  |  169  |  275  |
%C A256695 |  12240 |   12240  |   78  |  442  |  500  |
%C A256695 |  17136 |   92820  |  168  |  221  |  325  |
%C A256695 |  41580 |  183150  |  250  |  333  |  407  |
%C A256695 +--------+----------+-------+-------+-------+
%e A256695 a(1) = 12 because, for (a,b,c) = (5, 5, 8) => s = (5+5+8)/2 = 9, and
%e A256695 A = sqrt(9(9-5)(9-5)(9-8)) = sqrt(144) = 12 and the triangle (5+5, 5+8, 5+8) = (10, 13, 13)=> s1 = (10+13+13)/2 = 18, and A1 = sqrt(18(18-10)(18-13)(18-13)) = sqrt(3600) = 60 is an integer.
%t A256695 nn=1000;lst={};Do[s=(a+b+c)/2;If[IntegerQ[s],area2=s(s-a)(s-b)(s-c);u=a+b;v=a+c;w=b+c;s1=(u+v+w)/2;area3=s1(s1-u)(s1-v)(s1-w);If[area2>0&&area3>0&&IntegerQ[Sqrt[area2]]&&IntegerQ[Sqrt[area3]], AppendTo[lst,Sqrt[area2]]]],{a,nn},{b,a},{c,b}];Union[lst]
%Y A256695 Cf. A135453, A188158.
%K A256695 nonn
%O A256695 1,1
%A A256695 _Michel Lagneau_, Apr 08 2015