A253277 Integer area A of triangles with side lengths in the commutative ring Z[sqrt(3)].

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

Offset: 1

Author

Michel Lagneau, May 02 2015

Keywords

• nonn

Comments

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. |

+----+---------+----------+----------+-------+---------+

Links

• Eric Weisstein's World of Mathematics, Ring.

Crossrefs

Cf. A188158, A238369.

Programs

Mathematica

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]]]]