This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A254075 #23 Feb 16 2025 08:33:24
%S A254075 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,
%T A254075 33,35,36,38,40,42,44,45,48,49,50,52,54,55,57,58,60,63,64,65,66,72,75,
%U A254075 76,77,80,81,84,88,90,95,96,98,99,100,108,110,112,114,116
%N A254075 Integer area A of triangles with side lengths in the commutative ring Z[sqrt(5)].
%C A254075 Extension of A188158 with triangles of sides in the ring Z[sqrt(5)] = {x + y sqrt(5)| x,y in Z}.
%C A254075 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, ...}.
%C A254075 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.
%C A254075 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 A254075 Geometric property of the triangles in the ring Z[sqrt(5)]:
%C A254075 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 A254075 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.
%C A254075 Notation in the table:
%C A254075 q=sqrt(5)and irrat. = irrational numbers of the form u+v*q.
%C A254075 +----+---------+----------+----------+-------+---------+
%C A254075 | A | a | b | c | r | R |
%C A254075 +----+---------+----------+----------+-------+---------+
%C A254075 | 1 | 1 | 2 | q | irrat.| irrat. |
%C A254075 | 2 | 1 | 5 | 2q | irrat.| irrat. |
%C A254075 | 2 | 2 | q | q | irrat.| 5/4 |
%C A254075 | 2 | 4 | q | q | irrat.| 5/2 |
%C A254075 | 3 | 2 | 5 | 3q | irrat.| irrat. |
%C A254075 | 3 | 3 | q | 2q | irrat.| 5/2. |
%C A254075 | 4 | 1 | 17 | 8q | irrat.| irrat. |
%C A254075 | 4 | 2 | 4 | 2q | irrat.| irrat. |
%C A254075 | 5 | 2 | 13 | 5q | irrat.| irrat. |
%C A254075 | 5 | 5 | q | 2q | irrat.| 5/2 |
%C A254075 | 6 | 3 | 4 | 5 | 1 | 5/2 |
%C A254075 | 6 | 1 | 13 | 6q | irrat.| irrat. |
%C A254075 | 7 | 7 | 2q | 5q | irrat.| 25/2 |
%C A254075 | 8 | 2 | 10 | 4q | irrat.| irrat. |
%C A254075 | 8 | 4 | 2q | 2q | irrat.| 5/2 |
%C A254075 | 8 | 5 | 13 | 8q | irrat.| irrat. |
%C A254075 | 8 | 6 | 5-q | 5+q | 1 | 15/4 |
%C A254075 | 8 | 8 | 2q | 2q | irrat.| 5 |
%C A254075 +----+---------+----------+----------+-------+---------+
%H A254075 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Ring.html">Ring</a>
%t A254075 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]
%Y A254075 Cf. A188158, A238368, A238369, A253277.
%K A254075 nonn
%O A254075 1,2
%A A254075 _Michel Lagneau_, May 03 2015