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A305717 Number of distinct Heronian triangles with perimeter A051518(n).

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%I A305717 #27 Aug 12 2022 09:18:34
%S A305717 1,1,1,1,1,2,4,1,2,1,3,2,4,1,3,5,1,1,2,5,1,1,4,8,1,5,5,6,5,2,12,1,6,1,
%T A305717 5,4,9,1,9,2,5,11,8,2,3,5,7,12,1,10,1,1,1,7,10,3,2,2,15,14,5,10,5,5,
%U A305717 11,16,5,1,12,7,3,1,7,1,2,15,16,1,11,16,18
%N A305717 Number of distinct Heronian triangles with perimeter A051518(n).
%H A305717 Peter Kagey, <a href="/A305717/b305717.txt">Table of n, a(n) for n = 1..1000</a>
%H A305717 Mathematics Stack Exchange, <a href="https://math.stackexchange.com/q/2605938/121988">Properties of triangles with integer sides and area</a>
%H A305717 Wikipedia, <a href="https://en.wikipedia.org/wiki/Heronian_triangle">Heronian triangle</a>
%F A305717 a(n) = A051516(A051518(n)).
%e A305717 For n = 7, the a(7) = 4 Heronian triangles with perimeter A051518(7) = 36 are:
%e A305717 the 10-13-13 isosceles triangle (area 60),
%e A305717 the 10-10-16 isosceles triangle (area 48),
%e A305717 the  9-12-15 scalene   triangle (area 54), and
%e A305717 the  9-10-17 scalene   triangle (area 36).
%t A305717 htc[p_] := Block[{t=0, c, q=p/2}, Do[c=p-a-b; If[c >= b && a+c > b && a+b > c && IntegerQ[Sqrt[q (q-a) (q-b) (q-c)]], t++], {a, p/3}, {b, a, p-a-1}]; t]; Select[htc /@ (Range[128] 2), # > 0 &] (* _Giovanni Resta_, Jun 14 2018 *)
%Y A305717 Cf. A051516, A051518, A070138, A298614.
%K A305717 nonn
%O A305717 1,6
%A A305717 _Peter Kagey_, Jun 08 2018