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A070086 Areas of integer triangles [A070080(n), A070081(n), A070082(n)], rounded values.

%I A070086 #30 Jul 29 2025 20:04:45
%S A070086 0,1,2,1,2,3,2,3,4,4,4,2,4,4,6,5,6,7,3,5,5,7,8,6,7,8,9,3,6,6,9,7,10,
%T A070086 11,7,9,10,11,12,4,6,8,10,8,12,12,14,8,10,12,13,12,15,16,4,7,9,12,10,
%U A070086 14,10,15,16,17,9,12,13,15,14,17,18,19,5,8,10
%N A070086 Areas of integer triangles [A070080(n), A070081(n), A070082(n)], rounded values.
%C A070086 Triangles [A070080(A070142(n)), A070081(A070142(n)), A070082(A070142(n))] have integer areas = a(A070142(k)) = A070149(k).
%H A070086 Jean-François Alcover, <a href="/A070086/b070086.txt">Table of n, a(n) for n = 1..972</a>
%H A070086 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HeronsFormula.html">Heron's Formula</a>.
%H A070086 Reinhard Zumkeller, <a href="/A070080/a070080.txt">Integer-sided triangles</a>
%F A070086 a(n) = sqrt(s*(s-u)*(s-v)*(s-w)), where u=A070080(n), v=A070081(n), w=A070082(n) and s = A070083(n)/2 = (u+v+w)/2.
%e A070086 [A070080(25), A070081(25), A070082(25)] = [3,5,6] and s = A070083(25)/2 = (3+5+6)/2 = 7: a(25) = sqrt(s*(s-3)*(s-5)*(s-6)) = sqrt(7*(7-3)*(7-5)*(7-6)) = sqrt(7*4*2*1) = sqrt(56) = 7.48331, rounded = 7.
%t A070086 m = 50; (* max perimeter *)
%t A070086 sides[per_] := Select[Reverse /@ IntegerPartitions[per, {3}, Range[ Ceiling[per/2]]], #[[1]] < per/2 && #[[2]] < per/2 && #[[3]] < per/2 &];
%t A070086 triangles = DeleteCases[Table[sides[per], {per, 3, m}], {}] // Flatten[#, 1]& // SortBy[Total[#] m^3 + #[[1]] m^2 + #[[2]] m + #[[1]]&];
%t A070086 area[{a_, b_, c_}] := With[{p = (a+b+c)/2}, Sqrt[p(p-a)(p-b)(p-c)] // Round];
%t A070086 area /@ triangles (* _Jean-François Alcover_, Oct 03 2021 *)
%Y A070086 Cf. A051516, A055595, A046131.
%Y A070086 The sides are given by A070080, A070081, A070082.
%Y A070086 See A135622 for values signifying the precise area and further crossrefs.
%K A070086 nonn,easy
%O A070086 1,3
%A A070086 _Reinhard Zumkeller_, May 05 2002