A070150 Triangular areas of integer Heronian triangles.

6, 36, 66, 120, 36, 120, 120, 210, 210, 120, 300, 210, 210, 300, 378, 630, 528, 780, 528, 210, 630, 630, 300, 1176, 780, 2016, 990, 1176, 2016, 2016, 1596, 780, 1770, 528, 300, 2850, 630, 2016, 780, 990, 3240, 2016, 630

Offset: 1

Author

Reinhard Zumkeller, May 05 2002

Keywords

• nonn
• nice

Comments

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

Examples

			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).
		

Links

• Eric Weisstein's World of Mathematics, Heronian Triangle.
• R. Zumkeller, Integer-sided triangles

Programs

Mathematica

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 *)