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A055595 Area of triangles with integer sides and positive integer area, ordered by longest side, then second longest side and finally shortest side.

%I A055595 #14 Jul 29 2025 14:51:39
%S A055595 6,12,12,24,48,30,60,54,24,84,48,36,60,120,108,66,42,96,84,126,60,108,
%T A055595 192,90,150,84,168,120,36,204,240,210,210,60,120,216,132,300,96,336,
%U A055595 72,192,144,240,480,294,84,252,360,432,114,156,180,210,420,120,210,420
%N A055595 Area of triangles with integer sides and positive integer area, ordered by longest side, then second longest side and finally shortest side.
%C A055595 This is the ordering of triangles used for A316841.
%H A055595 Ray Chandler, <a href="/A055595/b055595.txt">Table of n, a(n) for n = 1..10000</a>
%H A055595 A. Dendane, <a href="http://www.analyzemath.com/Geometry_calculators/herons_area_triangle.html">Heron's Formula For Area of a Triangle-Geometry Calculator</a>
%F A055595 a(n) = sqrt(s(n)*(s(n)-A055592(n))*(s(n)-A055593(n))*(s(n)-A055594(n))) where s(n) = (A055592(n)+A055593(n)+A055594(n))/2 i.e. half the perimeter of the triangle
%t A055595 max = 42; triangles = Reap[Do[s = (a+b+c)/2; area = Sqrt[s*(s-a)*(s-b)*(s-c)]; If[IntegerQ[area] && area > 0, Sow[{a, b, c, area}]], {a, 1, max}, {b, a, max}, {c, b, max}]][[2, 1]]; A055595 = Sort[triangles, #1[[3]]*max^2 + #1[[2]]*max + #1[[1]] < #2[[3]]* max^2 + #2[[2]]*max + #2[[1]] &][[All, 4]](* _Jean-François Alcover_, Jun 12 2012 *)
%Y A055595 The sides are given by A055592, A055593, A055594.
%Y A055595 Cf. A046131, A070786, A316841, A316853.
%Y A055595 Range of values: A188158.
%K A055595 nonn
%O A055595 1,1
%A A055595 _Henry Bottomley_, May 26 2000