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

1,6

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

Formula

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

Example

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).

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 *)

Crossrefs

Cf. A051516, A051518, A070138, A298614.

Sequence in context: A030420 A133902 A265823 * A208645 A361381 A054708

Adjacent sequences: A305714 A305715 A305716 * A305718 A305719 A305720

Keyword

nonn

Author

Peter Kagey, Jun 08 2018

Status

approved