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%I #17 Sep 22 2013 14:10:18
%S 150,300,600,1050,1200,1350,2400,2700,3750,4200,4800,5070,5400,7350,
%T 7500,9450,9600,10140,10800,12150,14700,15000,16800,17340,18150,19200,
%U 20280,21600,24300,25350,26250,29400,30000,33750,34680,36300,37800
%N Areas A of the triangles such that A, the sides and the three altitudes are integers.
%C Properties of this sequence :
%C There exists three class of numbers included into a(n) :
%C (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.
%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).
%C (iii) A subset such that {1050, 4200, 9450,…} without the precedent properties.
%H Ray Chandler, <a href="/A210643/b210643.txt">Table of n, a(n) for n = 1..64</a>
%H Eric Weisstein, <a href="http://mathworld.wolfram.com/IsoscelesTriangle.html">Isosceles Triangle</a>
%H Eric Weisstein, <a href="http://mathworld.wolfram.com/Altitude.html">Altitude</a>
%e Primitive solutions follow:
%e Area, ( a, b, c), (h1, h2, h3), Case
%e 150, (15, 20, 25), (20, 15, 12), Right,
%e 300, (25, 25, 30), (24, 24, 20), Isosceles,
%e 300, (25, 25, 40), (24, 24, 15), Isosceles,
%e 1050, (35, 75, 100), (60, 28, 21), Other,
%e 5070, (65, 156, 169), (156, 65, 60), Right,
%e 10140, (130, 169, 169), (156, 120, 120), Isosceles,
%e 10140, (169, 169, 312), (120, 120, 65), Isosceles,
%e 17340, (136, 255, 289), (255, 136, 120), Right,
%e 34680, (272, 289, 289), (255, 240, 240), Isosceles,
%e 34680, (289, 289, 510), (240, 240, 136), Isosceles,
%e 52500, (175, 600, 625), (600, 175, 168), Right,
%e 82500, (275, 625, 750), (600, 264, 220), Other.
%p 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):
%t 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]
%Y Cf. A188158.
%K nonn
%O 1,1
%A _Michel Lagneau_, Mar 26 2012
%E More terms from _Ray Chandler_, Apr 24 2013
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