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A256695
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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.
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0
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12, 48, 108, 192, 300, 432, 588, 768, 972, 1008, 1200, 1452, 1728, 2028, 2352, 2448, 2520, 2700, 2772, 3060, 3072, 3468, 3888, 4032, 4332, 4800, 5292, 5808, 6348, 6912, 7500, 8112, 8748, 9072, 9408, 9792, 10080, 10092, 10800, 11088, 11532, 11628, 12240, 12288
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OFFSET
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1,1
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COMMENTS
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The areas of the primitive triangles are 12, 2520, 2772, 3060, 4032, 5808, 9072, 11088, 11628, 17136, 24948, 41580, ...
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.
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,...
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).
+-------+--------+------+------+------+
| A | A’ | a | b | c |
+-------+--------+------+------+------+
| 12 | 60 | 5 | 5 | 8 |
| 48 | 240 | 10 | 10 | 16 |
| 108 | 540 | 15 | 15 | 24 |
| 192 | 960 | 20 | 20 | 32 |
| 300 | 1500 | 25 | 25 | 40 |
| 432 | 2160 | 30 | 30 | 48 |
| 588 | 2940 | 35 | 35 | 56 |
| 768 | 3840 | 40 | 40 | 64 |
| 972 | 4860 | 45 | 45 | 72 |
+-------+--------+------+------+------+
We find a majority of isosceles triangles, but there is a subsequence of non-isosceles triangles with areas 2520, 3060, 10080, 11088, ...
+--------+----------+-------+-------+-------+
| A | A’ | a | b | c |
+--------+----------+-------+-------+-------+
| 2520 | 18270 | 29 | 174 | 175 |
| 3060 | 33150 | 39 | 221 | 250 |
| 10080 | 73080 | 58 | 348 | 350 |
| 11088 | 64350 | 150 | 169 | 275 |
| 12240 | 12240 | 78 | 442 | 500 |
| 17136 | 92820 | 168 | 221 | 325 |
| 41580 | 183150 | 250 | 333 | 407 |
+--------+----------+-------+-------+-------+
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LINKS
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EXAMPLE
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a(1) = 12 because, for (a,b,c) = (5, 5, 8) => s = (5+5+8)/2 = 9, and
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 integer.
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MATHEMATICA
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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].
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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