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A238369
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Integer area A of triangles with side lengths in the commutative ring Z[sqrt(2)].
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3
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1, 2, 3, 4, 6, 7, 8, 9, 10, 12, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 46, 48, 49, 50, 51, 52, 53, 54, 56, 60, 61, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 80, 81, 82, 84, 85
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OFFSET
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1,2
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COMMENTS
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Generalized integer areas triangles in the ring Z[sqrt(2)] = {a + b sqrt(2)| a,b in Z}.
The sequence A188158 is included in this sequence. The numbers 2*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(2), b*sqrt(2), c*sqrt(2)) is 2*A.
The primitive areas are 1, 3, 7, 9, 10, 15, 17, 19, 21, 25, ... and the numbers 2^p, 3*2^p, 7*2^p, ... are in the sequence. The numbers p^2*a(n) are in the sequence.
According to the limits of the Mathematica program, it is impossible to find integer areas of values 5, 11, 13, 22, 29, 39, 45, 47, 55, 57, 58, 59, 67, 71, 73, 78, 79, 83, 87, ... with sides in the ring Z(sqrt(2)).
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, for example the area of the triangles (3,4,5), (2,10,6*sqrt(2)),(3,6-sqrt(2),-3+5*sqrt(2)),(3,6+sqrt(2),3+5*sqrt(2)) and (7-4*sqrt(2), 3+7*sqrt(2), 4+7*sqrt(2)) is A = 6.
Geometric property of the triangles in the ring Z[sqrt(2)]
It is possible to obtain integers values (or rational values) for the irradius (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(2)] and r = A/p, R = a*b*c/(4*A) are respectively the values of the irradius and the circumradius.
Notation in the table:
q=sqrt(2) and irrat. = irrational numbers of the form u+v*q.
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| A | a | b | c | r | R |
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| 1 | q | q | 2 | irrat.| 1 |
| 2 | 1 | 5 | 4*q | irrat.| irrat. |
| 3 | 6 | q | 5*q | irrat.| 5 |
| 4 | 6 | 5-2*q | 5+2*q | 1/2 | 51/8 |
| 6 | 3 | 4 | 5 | 1 | 5/2 |
| 7 | 2 | 5*q | 5*q | irrat.| irrat. |
| 8 | 4 | 4 | 4*q | irrat.| irrat. |
| 9 | 6 | 3*q | 3*q | irrat.| 6 |
| 10 | 5*q | 9-2*q | -1+3*q | irrat.| irrat. |
| 12 | 5 | 5 | 6 | 3/2 | 25/8 |
| 14 | 5 | 7 | 4*q | irrat.| irrat. |
| 15 | 10 | -4+5*q | 4+5*q | irrat.| 17/3 |
| 16 | 8 | 4*q | 4*q | irrat.| 4 |
| 17 | 18 | -8+7*q | 8+7*q | irrat.| 9 |
| 18 | 6 | 6 | 6*q | 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=40; q=Sqrt[2]; 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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