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 A256584 #9 Feb 16 2025 08:33:25
%S A256584 6,12,54,96,108,126,144,180,192,216,234,240,264,270,336,360,378,408,
%T A256584 480,486,504,522,540,594,600,744,750,756,864,900,972,990,1008,1026,
%U A256584 1116,1134,1224,1296,1350,1386,1404,1494,1500,1536,1584,1620,1656,1728,1800,1872
%N A256584 Integer areas of integer-sided triangles where at least one of the three altitudes is of perfect square length.
%C A256584 a(n) contains A210643.
%C A256584 There are triangles with rational square, for instance, with the area 144, we find for (a,b,c)=(6,50,52) the altitudes {Ha,Hb,Hc} = {72/13, 144/25, 48} but with the same area we find also for (a,b,c)=(18,20,34) the altitudes {Ha,Hb,Hc} = {144/17, 72/5, 16}.
%C A256584 The corresponding squares are 4, 4, 9, 16, 9, 9, 16, 9, 16, 36, 9, 16, 16, 36, 16, 9, 36, 16, 16, 36, 16, 36, 36, 36, 16, 16, 25, 36, 36, 36, 36, 36, 16, 36, 36, 36, 144, 144, 36, 36, 36, 36, 100, 64, 144, 36, 36, 36, 144, 36, ...
%C A256584 The subsequence of the primitive triangles are 6, 12, 126, 144, 180, 216, 234, ...
%C A256584 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.
%C A256584 The altitudes of a triangle with sides length a, b, c and area A have length given by Ha= 2A/a, Hb= 2A/b, Hc= 2A/c.
%C A256584 Properties of this sequence:
%C A256584 - The sequence is infinite because from de initial primitive triangle (3,4,5), the area A’ of the triangle (3*3^m, 4*3^m, 5*3^m) is also in the sequence where A’ = 6*3^2m and {Ha, Hb, Hc} = {4*3^m, 3^(m+1), (4*3^(m+1))/5}. The altitude Ha or Hb is square.
%C A256584 - There are three subsets of numbers included into a(n):
%C A256584 Case (i): A subset with right triangles (a,b,c) where a^2+b^2 = c^2 with area a2(n) = {6, 54, 96, 180, 240, 270, ...}
%C A256584 Case (ii): A subset with isosceles triangles of area a1(n)= {12, 108, 192, 360, 480, 540, ...} = 2*a1(n).
%C A256584 Case (iii): A subset with non-isosceles and non-right triangles of area a3(n)= {126, 144, 216, 234, 264, 336, ...}
%C A256584 -----------------------------------------------------------
%C A256584 | A | a | b | c | Ha | Hb | Hc |
%C A256584 -----------------------------------------------------------
%C A256584 | 6 | 3 | 4 | 5 | 12/5 | 3 | 4 |
%C A256584 | 12 | 5 | 5 | 6 | 4 | 24/5 | 24/5 |
%C A256584 | 54 | 9 | 12 | 15 | 36/5 | 9 | 12 |
%C A256584 | 96 | 12 | 16 | 20 | 48/5 | 12 | 16 |
%C A256584 | 108 | 15 | 15 | 24 | 9 | 72/5 | 72/5 |
%C A256584 | 126 | 15 | 28 | 41 | 252/41 | 9 | 84/5 |
%C A256584 | 144 | 18 | 20 | 34 | 144/17 | 72/5 | 16 |
%C A256584 | 180 | 9 | 40 | 41 | 360/41 | 9 | 40 |
%C A256584 | 192 | 20 | 20 | 24 | 16 | 96/5 | 96/5 |
%C A256584 | 216 | 12 | 39 | 45 | 48/5 | 144/13 | 36 |
%C A256584 | 234 | 15 | 41 | 52 | 9 | 468/41 | 156/5 |
%C A256584 | 240 | 16 | 30 | 34 | 240/17 | 16 | 30 |
%C A256584 | 264 | 33 | 34 | 65 | 528/65 | 264/17 | 16 |
%H A256584 Eric Weisstein, <a href="https://mathworld.wolfram.com/Altitude.html">Altitude</a>
%H A256584 Eric Weisstein, <a href="https://mathworld.wolfram.com/IsoscelesTriangle.html">Isosceles Triangle</a>
%H A256584 Eric Weisstein, <a href="https://mathworld.wolfram.com/RightTriangle.html">Right Triangle</a>
%e A256584 1350 is in the sequence because the altitudes of the triangle (45, 60, 75) are (60, 45, 36).
%t A256584 nn=200;lst={};Do[s=(a+b+c)/2;If[IntegerQ[s],area2=s (s-a) (s-b) (s-c);If[0<area2&&IntegerQ[Sqrt[area2]]&&(IntegerQ[Sqrt[2*Sqrt[area2]/a]]||IntegerQ[Sqrt[2*Sqrt[area2]/b]]||IntegerQ[Sqrt[2*Sqrt[area2]/c]]),AppendTo[lst,Sqrt[area2]]]],{a,nn},{b,a},{c,b}];Union[lst]
%Y A256584 Cf. A210643.
%K A256584 nonn
%O A256584 1,1
%A A256584 _Michel Lagneau_, Apr 02 2015