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

%I A253277 #19 Feb 16 2025 08:33:24
%S A253277 3,6,9,12,18,21,24,27,30,33,36,39,42,48,49,54,60,63,66,72,75,78,81,84,
%T A253277 90,96,99,108,114,117,120,126,132,138,144,147,150,156,162,168,180,189,
%U A253277 192,196,198,204,210,216,222,225,227,228,234,240,243,252,264,270
%N A253277 Integer area A of triangles with side lengths in the commutative ring Z[sqrt(3)].
%C A253277 Extension of A188158 with triangles of sides in the ring Z[sqrt(3)] = {a + b sqrt(3)| a,b in Z}.
%C A253277 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.
%C A253277 The squares of the sequence are 9, 36, 49, 81, 144, 196, 225, ...
%C A253277 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.
%C A253277 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).
%C A253277 Geometric property of the triangles in the ring Z[sqrt(3)]:
%C A253277 It is possible to obtain integers values (or rational values) for the inradius (and/or) the circumradius of the triangles (see the table below).
%C A253277 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.
%C A253277 Notation in the table:
%C A253277 q=sqrt(3)and irrat. = irrational numbers of the form u+v*q.
%C A253277 +----+---------+----------+----------+-------+---------+
%C A253277 |  A |    a    |      b   |     c    |   r   |  R      |
%C A253277 +----+---------+----------+----------+-------+---------+
%C A253277 |  3 |   3 - q |  2 + 2q  |  1 + 3q  | irrat.|  irrat. |
%C A253277 |  3 |   3 + q | -2 + 2q  | -1 + 3q  | irrat.|  irrat. |
%C A253277 |  6 |   3     |  4       |  5       |   1   |   5/2   |
%C A253277 |  6 |   8     |  5 - 2q  |  5 + 2q  | 2/3   |  13/3   |
%C A253277 |  6 |  4q     |  4 - q   |  4 + q   | irrat.|  irrat. |
%C A253277 |  6 |  8q     |  7 - 2q  |  7 + 2q  | irrat.|  irrat. |
%C A253277 |  9 |  3 + 3q |  6 - 2q  |  9 - q   |   1   |  irrat. |
%C A253277 | 12 |   5     |  5       |  6       |  3/2  |  25/8   |
%C A253277 | 12 |   5     |  5       |  8       |  4/3  |  25/6   |
%C A253277 | 12 |  2q     | -1 + 5q  |  1 + 5q  | irrat.|  irrat. |
%C A253277 | 12 |   6     | -1 + 3q  |  1 + 3q  | irrat.|  13/4   |
%C A253277 | 18 |  12     | -3 + 4q  |  3 + 4q  | irrat.|  13/2   |
%C A253277 | 21 |  9 + q  | -2 + 6q  | -7 + 7q  | irrat.|  irrat. |
%C A253277 +----+---------+----------+----------+-------+---------+
%H A253277 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Ring.html">Ring</a>.
%t A253277 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]]]]<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]
%Y A253277 Cf. A188158, A238369.
%K A253277 nonn
%O A253277 1,1
%A A253277 _Michel Lagneau_, May 02 2015