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

1,1

Comments

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

Links

Table of n, a(n) for n=1..43.

Eric Weisstein's World of Mathematics, Heronian Triangle.

R. Zumkeller, Integer-sided triangles

Example

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).

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 *)

Crossrefs

Sequence in context: A134639 A069497 A139249 * A161144 A364586 A119845

Adjacent sequences: A070147 A070148 A070149 * A070151 A070152 A070153

Keyword

nonn,nice

Author

Reinhard Zumkeller, May 05 2002

Status

approved