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

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, 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, 11, 16, 5, 1, 12, 7, 3, 1, 7, 1, 2, 15, 16, 1, 11, 16, 18

Offset: 1

Author

Peter Kagey, Jun 08 2018

Keywords

• nonn

Examples

			For n = 7, the a(7) = 4 Heronian triangles with perimeter A051518(7) = 36 are:
the 10-13-13 isosceles triangle (area 60),
the 10-10-16 isosceles triangle (area 48),
the  9-12-15 scalene   triangle (area 54), and
the  9-10-17 scalene   triangle (area 36).
		

Links

• Peter Kagey, Table of n, a(n) for n = 1..1000
• Mathematics Stack Exchange, Properties of triangles with integer sides and area
• Wikipedia, Heronian triangle

Crossrefs

Cf. A051516, A051518, A070138, A298614.

Programs

Mathematica

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 *)

Formula

a(n) = A051516(A051518(n)).