A210643 Areas A of the triangles such that A, the sides and the three altitudes are integers.

150, 300, 600, 1050, 1200, 1350, 2400, 2700, 3750, 4200, 4800, 5070, 5400, 7350, 7500, 9450, 9600, 10140, 10800, 12150, 14700, 15000, 16800, 17340, 18150, 19200, 20280, 21600, 24300, 25350, 26250, 29400, 30000, 33750, 34680, 36300, 37800

Offset: 1

Author

Michel Lagneau, Mar 26 2012

Keywords

• nonn

Comments

Properties of this sequence :

There exists three class of numbers included into a(n) :

(i) A subset such that {150, 600, 1350, 2400, 3750, 5070,…} where the sides a h1 = b, h2 = a, h3 = a*b/c.

(ii) A subset such that a(n) = 300*n^2 = {300, 1200, 2700, 4800, …} where the triangles (a,b,c) are isosceles with a = b < c, and it is easy to check that a = b = 25*n, c=30*n, h1 = h2 = 24*n and h3 = sqrt(b^2 - c^2/4).

(iii) A subset such that {1050, 4200, 9450,…} without the precedent properties.

Examples

			Primitive solutions follow:
Area,  ( a,   b,   c),  (h1,  h2,  h3), Case
  150,  (15,  20,  25),  (20,  15,  12), Right,
  300,  (25,  25,  30),  (24,  24,  20), Isosceles,
  300,  (25,  25,  40),  (24,  24,  15), Isosceles,
1050,  (35,  75, 100),  (60,  28,  21), Other,
5070,  (65, 156, 169), (156,  65,  60), Right,
10140, (130, 169, 169), (156, 120, 120), Isosceles,
10140, (169, 169, 312), (120, 120,  65), Isosceles,
17340, (136, 255, 289), (255, 136, 120), Right,
34680, (272, 289, 289), (255, 240, 240), Isosceles,
34680, (289, 289, 510), (240, 240, 136), Isosceles,
52500, (175, 600, 625), (600, 175, 168), Right,
82500, (275, 625, 750), (600, 264, 220), Other.
		

Links

• Ray Chandler, Table of n, a(n) for n = 1..64
• Eric Weisstein, Isosceles Triangle
• Eric Weisstein, Altitude

Crossrefs

Cf. A188158.

Programs

Maple

with(numtheory):T:=array(1..1000):k:=0:nn:=500:for a from 1 to nn do: for b from a to nn do: for c from b to nn do:p:=(a+b+c)/2:s:=p*(p-a)*(p-b)*(p-c):if s>0 then s1:=sqrt(s): h1:=2*s1/a: h2:=2*s1/b:h3:=2*s1/c:if s1=floor(s1) and h1=floor(h1) and h2=floor(h2) and h3=floor(h3) then k:=k+1:T[k]:=s1:else fi:fi:od:od:od: L := [seq(T[i],i=1..k)]:L1:=convert(T,set):A:=sort(L1, `<`): print(A):

Mathematica

nn = 900; lst = {}; Do[s = (a + b + c)/2; If[IntegerQ[s], area2 = s (s - a) (s - b) (s - c); If[0 < area2 <= nn^2 && IntegerQ[Sqrt[area2]]&&IntegerQ[(2*Sqrt[area2])/a] &&IntegerQ[(2*Sqrt[area2])/b] &&IntegerQ[(2*Sqrt[area2])/c], AppendTo[lst, Sqrt[area2]]]], {a, nn}, {b, a}, {c, b}]; Union[lst]

Extensions

More terms from Ray Chandler, Apr 24 2013