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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. |

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

Generalized integer areas triangles in the ring Z[sqrt(q)] = {a + b sqrt(q)| a,b in Z}.

Introduction:

The study of triangles having their sides with values in a ring Z[sqrt(q)] and having integer area gives remarkable properties probably still unexplored today.

Property:

a(1) = 6 because the ring Z[sqrt(1)] = Z => the smallest area of integer sides is A188158(1) = 6 => a(n) <=6.

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 instance, in the ring Z[sqrt(19)] the area of each following triangle:

(3, 4, 5),

(8, -2+sqrt(19), 2+sqrt(19)),

(6-sqrt(q), 3+sqrt(19), -1+2*sqrt(19)),

(-3+sqrt(19), 6+sqrt(19), 1+2*sqrt(19)) is A=6.

Conjecture: the set of squarefree numbers q such that the integer area A of the triangles with sides in the commutative ring Z[sqrt(q)] is finite if A < 6.

It follows that a(n)= 6 for n > 384 where A005117(384) = 629 and a(384)=5.

The corresponding values q such that a(n)<6 are 2, 3, 5, 7, 10, 11, 13, 14, 15, 17, 26, 29, 37, 41, 65, 85, 89, 101, 113, 221 and 629 with the corresponding index in the sequence a(n): 2, 3, 4, 6, 7, 8, 9, 10, 11, 12, 17, 18, 24, 27, 40, 53, 56, 62, 71, 137 and 384.

We observe a subset of numbers q such that the sides of the triangles are of the particular form (a, b, sqrt(q)) with a, b integers. This subset is {5, 13, 17, 29, 37, 41, 65, 85, 89, 101, 113, 221, 629}. See the table below for the examples.

We find also five isosceles triangles with area less than 6 and it is observed that they are of the form (2,sqrt(q),sqrt(q)) with q = 2, 5, 10, 17 and 26. The corresponding areas A are 1, 2, 3, 4 and 5 respectively with the formula A = sqrt(q-1).

The following table gives the first values (A, sqrt(q), a, b, c) where A is the smallest area of the triangle (a, b, c), Z[sqrt(q)] is the commutative ring and a, b, c are the sides in Z[sqrt(q)]= Z[sqrt(A005117(n))].

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

| A | sqrt(q) | a | b | c |

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

| 6 | sqrt(1) | 2 | 3 | 4 |

| 1 | sqrt(2) | 2 | sqrt(2) | sqrt(2) |

| 3 | sqrt(3) | 3 - sqrt(3) | 2 + 2*sqrt(3)| 1 + 3*sqrt(3) |

| 1 | sqrt(5) | 1 | 2 | sqrt(5) |

| 6 | sqrt(6) | 2*sqrt(6) |-2 + 2*sqrt(6)| 2 + 2*sqrt(6) |

| 3 | sqrt(7) | 4 | -1 + sqrt(7) | 1 + sqrt(7) |

| 3 | sqrt(10) | 2 | sqrt(10) | sqrt(10) |

| 4 | sqrt(11) | 6 |-1 + sqrt(11) | 1 + sqrt(11) |

| 3 | sqrt(13) | 2 | 3 | sqrt(13) |

| 5 | sqrt(14) | 6 |-2 + sqrt(14) | 2 + sqrt(14) |

| 3 | sqrt(15) | 8 | 5 - sqrt(15) | 5 + sqrt(15) |

| 2 | sqrt(17) | 1 | 4 | sqrt(17) |

| 6 | sqrt(19) | 8 |-2 + sqrt(19) | 2 + sqrt(19) |

| 6 | sqrt(21) | 3 | 4 | 5 |

| 6 | sqrt(22) | 3 | 4 | 5 |

| 6 | sqrt(23) | 3 | 4 | 5 |

| 3 | sqrt(26) | 10 |-4 + sqrt(26) | 4 + sqrt(26) |

| 5 | sqrt(29) | 2 | 5 | sqrt(29) |

| 6 | sqrt(30) | 3 | 4 | 5 |

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

Extension of A188158 with triangles of sides in the ring Z[sqrt(5)] = {x + y sqrt(5)| x,y in Z}.

The numbers 5*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(5), b*sqrt(5), c*sqrt(5)) is 5*A. The numbers a(n)*5^p and a(n)*q^2 are in the sequence. Because a(1)=1, the squares are in the sequence. The primitive areas of the sequence are {1, 2, 3, 6, 7, 11, 13, 19, ...}.

The values shown were obtained with a and b in the range [-40, ..., +40]. For the areas > 120 it would be necessary to expand the range of variation, but then the calculations would become very slow.

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(5)]:

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(5)] 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(5)and irrat. = irrational numbers of the form u+v*q.

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

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

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

| 1 | 1 | 2 | q | irrat.| irrat. |

| 2 | 1 | 5 | 2q | irrat.| irrat. |

| 2 | 2 | q | q | irrat.| 5/4 |

| 2 | 4 | q | q | irrat.| 5/2 |

| 3 | 2 | 5 | 3q | irrat.| irrat. |

| 3 | 3 | q | 2q | irrat.| 5/2. |

| 4 | 1 | 17 | 8q | irrat.| irrat. |

| 4 | 2 | 4 | 2q | irrat.| irrat. |

| 5 | 2 | 13 | 5q | irrat.| irrat. |

| 5 | 5 | q | 2q | irrat.| 5/2 |

| 6 | 3 | 4 | 5 | 1 | 5/2 |

| 6 | 1 | 13 | 6q | irrat.| irrat. |

| 7 | 7 | 2q | 5q | irrat.| 25/2 |

| 8 | 2 | 10 | 4q | irrat.| irrat. |

| 8 | 4 | 2q | 2q | irrat.| 5/2 |

| 8 | 5 | 13 | 8q | irrat.| irrat. |

| 8 | 6 | 5-q | 5+q | 1 | 15/4 |

| 8 | 8 | 2q | 2q | irrat.| 5 |

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