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A070150 Triangular areas of integer Heronian triangles. 0
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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

LINKS

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

R. Zumkeller, Integer-sided triangles

Eric Weisstein's World of Mathematics, Heronian Triangle.

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-Fran├žois Alcover, Feb 22 2013 *)

CROSSREFS

Sequence in context: A134639 A069497 A139249 * A161144 A119845 A208064

Adjacent sequences:  A070147 A070148 A070149 * A070151 A070152 A070153

KEYWORD

nonn,nice

AUTHOR

Reinhard Zumkeller, May 05 2002

STATUS

approved

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Last modified November 21 12:45 EST 2014. Contains 249778 sequences.