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A238369 Integer area A of triangles with side lengths in the commutative ring Z[sqrt(2)]. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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.

---------------------------------------------------------

|  A |   a     |      b      |      c |   r   |  R      |

---------------------------------------------------------

|  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. |

........................................................

LINKS

Table of n, a(n) for n=1..67.

Wolfram MathWorld, Ring

MATHEMATICA

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]

CROSSREFS

Cf. A188158.

Sequence in context: A285598 A275804 A141825 * A296858 A296241 A070932

Adjacent sequences:  A238366 A238367 A238368 * A238370 A238371 A238372

KEYWORD

nonn

AUTHOR

Michel Lagneau, Feb 25 2014

STATUS

approved

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Last modified February 25 08:48 EST 2020. Contains 332221 sequences. (Running on oeis4.)