A070086 - OEIS

%I #25 Oct 04 2021 07:23:32

%S 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 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 14,10,15,16,17,9,12,13,15,14,17,18,19,5,8,10

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

%C Triangles [A070080(A070142(n)), A070081(A070142(n)), A070082(A070142(n))] have integer areas = a(A070142(k)) = A070149(k).

%H Jean-François Alcover, <a href="/A070086/b070086.txt">Table of n, a(n) for n = 1..972</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HeronsFormula.html">Heron's Formula</a>.

%H Reinhard Zumkeller, <a href="/A070080/a070080.txt">Integer-sided triangles</a>

%F 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 [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 m = 50; (* max perimeter *)

%t sides[per_] := Select[Reverse /@ IntegerPartitions[per, {3}, Range[ Ceiling[per/2]]], #[[1]] < per/2 && #[[2]] < per/2 && #[[3]] < per/2 &];

%t triangles = DeleteCases[Table[sides[per], {per, 3, m}], {}] // Flatten[#, 1]& // SortBy[Total[#] m^3 + #[[1]] m^2 + #[[2]] m + #[[1]]&];

%t area[{a_, b_, c_}] := With[{p = (a+b+c)/2}, Sqrt[p(p-a)(p-b)(p-c)] // Round];

%t area /@ triangles (* _Jean-François Alcover_, Oct 03 2021 *)

%Y Cf. A051516, A055595, A069596, A069594, A046131.

%K nonn

%O 1,3

%A _Reinhard Zumkeller_, May 05 2002