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A253277
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Integer area A of triangles with side lengths in the commutative ring Z[sqrt(3)].
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1
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3, 6, 9, 12, 18, 21, 24, 27, 30, 33, 36, 39, 42, 48, 49, 54, 60, 63, 66, 72, 75, 78, 81, 84, 90, 96, 99, 108, 114, 117, 120, 126, 132, 138, 144, 147, 150, 156, 162, 168, 180, 189, 192, 196, 198, 204, 210, 216, 222, 225, 227, 228, 234, 240, 243, 252, 264, 270
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OFFSET
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1,1
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COMMENTS
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Extension of A188158 with triangles of sides in the ring Z[sqrt(3)] = {a + b sqrt(3)| a,b in Z}.
The numbers 3*A188158(n) are in the sequence because if the integer area of the integer-sided triangle (a, b, c) is A, the area of the triangle of sides (a*sqrt(3), b*sqrt(3), c*sqrt(3)) is 3*A. The primitive areas of the sequence are in the subsequence b(n)={3, 6, 21, 30, 33, 39, 42, 49, ...} => the numbers b(n)*3^p and b(n)*q^2 are in the sequence.
The squares of the sequence are 9, 36, 49, 81, 144, 196, 225, ...
This sequence is tested with a and b in the range [-40, ..., +40]. For the values of areas > 400 it is necessary to expand the range of variation, but nevertheless the calculations become very long.
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. For the same area, the number of triangles is not unique (see the table below).
Geometric property of the triangles in the ring Z[sqrt(3)]:
It is possible to obtain integers values (or rational values) for the inradius (and/or) the circumradius of the triangles (see the table below).
The following table gives the first values (A, a, b, c, r, R) where A is the integer area, a,b,c are the sides in Z[sqrt(3)] and r = A/p, R = a*b*c/(4*A) are the values of the inradius and the circumradius respectively.
Notation in the table:
q=sqrt(3)and irrat. = irrational numbers of the form u+v*q.
+----+---------+----------+----------+-------+---------+
| A | a | b | c | r | R |
+----+---------+----------+----------+-------+---------+
| 3 | 3 - q | 2 + 2q | 1 + 3q | irrat.| irrat. |
| 3 | 3 + q | -2 + 2q | -1 + 3q | irrat.| irrat. |
| 6 | 3 | 4 | 5 | 1 | 5/2 |
| 6 | 8 | 5 - 2q | 5 + 2q | 2/3 | 13/3 |
| 6 | 4q | 4 - q | 4 + q | irrat.| irrat. |
| 6 | 8q | 7 - 2q | 7 + 2q | irrat.| irrat. |
| 9 | 3 + 3q | 6 - 2q | 9 - q | 1 | irrat. |
| 12 | 5 | 5 | 6 | 3/2 | 25/8 |
| 12 | 5 | 5 | 8 | 4/3 | 25/6 |
| 12 | 2q | -1 + 5q | 1 + 5q | irrat.| irrat. |
| 12 | 6 | -1 + 3q | 1 + 3q | irrat.| 13/4 |
| 18 | 12 | -3 + 4q | 3 + 4q | irrat.| 13/2 |
| 21 | 9 + q | -2 + 6q | -7 + 7q | irrat.| irrat. |
+----+---------+----------+----------+-------+---------+
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LINKS
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Eric Weisstein's World of Mathematics, Ring.
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MATHEMATICA
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err=1/10^10; nn=10; q=Sqrt[3]; lst={}; lst1={}; Do[If[u+q*v>0, lst=Union[lst, {u+q*v}]], {u, -nn, nn}, {v, -nn, nn}]; n1=Length[lst]; Do[a=Part[lst, i]; b=Part[lst, j]; c=Part[lst, k]; s=(a+b+c)/2; area2=s*(s-a)*(s-b)*(s-c); If[a*b*c !=0&&N[area2]>0&&Abs[N[Sqrt[area2]]-Round[N[Sqrt[area2]]]]<err, AppendTo[lst1, Round[Sqrt[N[area2]]]]; Print[Round[Sqrt[N[area2]]], " ", a, " ", b, " ", c]], {i, 1, n1}, {j, i, n1}, {k, j, n1}]; Union[lst1]
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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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