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 A238369 #13 Feb 16 2025 08:33:21
%S A238369 1,2,3,4,6,7,8,9,10,12,14,15,16,17,18,19,20,21,23,24,25,26,27,28,30,
%T A238369 31,32,33,34,35,36,37,38,40,41,42,43,44,46,48,49,50,51,52,53,54,56,60,
%U A238369 61,62,63,64,65,66,68,69,70,72,74,75,76,77,80,81,82,84,85
%N A238369 Integer area A of triangles with side lengths in the commutative ring Z[sqrt(2)].
%C A238369 Generalized integer areas triangles in the ring Z[sqrt(2)] = {a + b sqrt(2)| a,b in Z}.
%C A238369 The sequence A188158 is included in this sequence. The numbers 2*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(2), b*sqrt(2), c*sqrt(2)) is 2*A.
%C A238369 The primitive areas are 1, 3, 7, 9, 10, 15, 17, 19, 21, 25, ... and the numbers 2^p, 3*2^p, 7*2^p, ... are in the sequence. The numbers p^2*a(n) are in the sequence.
%C A238369 According to the limits of the Mathematica program, it is impossible to find integer areas of values 5, 11, 13, 22, 29, 39, 45, 47, 55, 57, 58, 59, 67, 71, 73, 78, 79, 83, 87, ... with sides in the ring Z(sqrt(2)).
%C A238369 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 example the area of the triangles (3,4,5), (2,10,6*sqrt(2)),(3,6-sqrt(2),-3+5*sqrt(2)),(3,6+sqrt(2),3+5*sqrt(2)) and (7-4*sqrt(2), 3+7*sqrt(2), 4+7*sqrt(2)) is A = 6.
%C A238369 Geometric property of the triangles in the ring Z[sqrt(2)]
%C A238369 It is possible to obtain integers values (or rational values) for the irradius (and/or) the circumradius of the triangles (see the table below).
%C A238369 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(2)] and r = A/p, R = a*b*c/(4*A) are respectively the values of the irradius and the circumradius.
%C A238369 Notation in the table:
%C A238369 q=sqrt(2) and irrat. = irrational numbers of the form u+v*q.
%C A238369 ---------------------------------------------------------
%C A238369 | A | a | b | c | r | R |
%C A238369 ---------------------------------------------------------
%C A238369 | 1 | q | q | 2 | irrat.| 1 |
%C A238369 | 2 | 1 | 5 | 4*q | irrat.| irrat. |
%C A238369 | 3 | 6 | q | 5*q | irrat.| 5 |
%C A238369 | 4 | 6 | 5-2*q | 5+2*q | 1/2 | 51/8 |
%C A238369 | 6 | 3 | 4 | 5 | 1 | 5/2 |
%C A238369 | 7 | 2 | 5*q | 5*q | irrat.| irrat. |
%C A238369 | 8 | 4 | 4 | 4*q | irrat.| irrat. |
%C A238369 | 9 | 6 | 3*q | 3*q | irrat.| 6 |
%C A238369 | 10 | 5*q | 9-2*q | -1+3*q | irrat.| irrat. |
%C A238369 | 12 | 5 | 5 | 6 | 3/2 | 25/8 |
%C A238369 | 14 | 5 | 7 | 4*q | irrat.| irrat. |
%C A238369 | 15 | 10 | -4+5*q | 4+5*q | irrat.| 17/3 |
%C A238369 | 16 | 8 | 4*q | 4*q | irrat.| 4 |
%C A238369 | 17 | 18 | -8+7*q | 8+7*q | irrat.| 9 |
%C A238369 | 18 | 6 | 6 | 6*q | irrat.| irrat. |
%C A238369 ........................................................
%H A238369 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Ring.html">Ring</a>
%t A238369 err=1/10^10;nn=40;q=Sqrt[2];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 A238369 Cf. A188158.
%K A238369 nonn
%O A238369 1,2
%A A238369 _Michel Lagneau_, Feb 25 2014