A070150 - OEIS

%I #12 Jun 12 2017 09:47:34

%S 6,36,66,120,36,120,120,210,210,120,300,210,210,300,378,630,528,780,

%T 528,210,630,630,300,1176,780,2016,990,1176,2016,2016,1596,780,1770,

%U 528,300,2850,630,2016,780,990,3240,2016,630

%N Triangular areas of integer Heronian triangles.

%C a(n) = A070086(A070148(n)).

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HeronianTriangle.html">Heronian Triangle</a>.

%H R. Zumkeller, <a href="/A070080/a070080.txt">Integer-sided triangles</a>

%e A070148(2)=368: [A070080(368), A070081(368), A070082(368)] = [9,10,17], area^2 = s*(s-9)*(s-10)*(s-17) with s=A070083(368)/2=(9+10+17)/2=18, area^2=18*9*8*1=16*81 is an integer square, therefore area=4*9=36=A000217(8).

%t maxPerim = 300; maxSide = Floor[(maxPerim - 1)/2]; order[{a_, b_, c_}] := (a + b + c)*maxPerim^3 + a*maxPerim^2 + b*maxPerim + c; triangles = Reap[ Do[ If[ a + b + c <= maxPerim && c - b < a < c + b && b - a < c < b + a && c - a < b < c + a, Sow[{a, b, c}]], {a, 1, maxSide}, {b, a, maxSide}, {c, b, maxSide}]][[2, 1]]; stri = Sort[ triangles, order[#1] < order[#2] &]; area[{a_, b_, c_}] := With[{p = (a + b + c)/2}, Sqrt[p*(p - a)*(p - b)*(p - c)]]; triangularQ[n_] := IntegerQ[Sqrt[8*n + 1]]; area /@ Select[stri, IntegerQ[area[#]] && triangularQ[area[#]] &] (* _Jean-François Alcover_, Feb 22 2013 *)

%K nonn,nice

%O 1,1

%A _Reinhard Zumkeller_, May 05 2002