A254075 Integer area A of triangles with side lengths in the commutative ring Z[sqrt(5)].

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 18, 19, 20, 22, 24, 25, 27, 28, 29, 30, 32, 33, 35, 36, 38, 40, 42, 44, 45, 48, 49, 50, 52, 54, 55, 57, 58, 60, 63, 64, 65, 66, 72, 75, 76, 77, 80, 81, 84, 88, 90, 95, 96, 98, 99, 100, 108, 110, 112, 114, 116

Offset

1,2

Comments

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 |

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

Links

Table of n, a(n) for n=1..66.

Eric Weisstein's World of Mathematics, Ring

Mathematica

err=1/10^10; nn=40; q=Sqrt[5]; 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, A238368, A238369, A253277.

Sequence in context: A367613 A095392 A232528 * A140401 A272916 A138389

Adjacent sequences: A254072 A254073 A254074 * A254076 A254077 A254078

Keyword

nonn

Author

Michel Lagneau, May 03 2015

Status

approved