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

1,1

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<b<c have the property a^2 + b^2 = c^2 => 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.

Links

Ray Chandler, Table of n, a(n) for n = 1..64

Eric Weisstein, Isosceles Triangle

Eric Weisstein, Altitude

Example

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.

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]

Crossrefs

Cf. A188158.

Sequence in context: A335145 A063829 A291959 * A211550 A212465 A273322

Adjacent sequences: A210640 A210641 A210642 * A210644 A210645 A210646

Keyword

nonn

Author

Michel Lagneau, Mar 26 2012

Extensions

More terms from Ray Chandler, Apr 24 2013

Status

approved