A051517 Triangles of perimeter 2n having integer sides and area.

0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 2, 0, 4, 0, 1, 2, 1, 0, 3, 2, 0, 4, 1, 0, 3, 0, 5, 1, 1, 2, 5, 0, 1, 1, 4, 0, 8, 0, 1, 5, 0, 0, 5, 6, 5, 0, 2, 0, 12, 1, 6, 1, 0, 0, 5, 0, 0, 4, 9, 1, 9, 0, 2, 0, 5, 0, 11, 0, 0, 8, 2, 3, 5, 0, 7, 12, 1, 0, 10, 1, 1, 1, 7, 0, 10, 3, 0, 2, 0, 2, 15, 0, 14, 5, 10, 0, 5, 0, 5, 11, 0, 0

Offset

1,16

Comments

No such triangles with odd perimeter exist.

Links

Frank M Jackson, Table of n, a(n) for n = 1..10000

Alan F. Beardon and Paul Stephenson, The Heron parameters of a triangle, Mathematical Gazette, Vol. 99, No. 545 (2015), pp. 205-212.

Example

a(18) = 4 because there are 4 triangles that have integer sides and area (Heronian triangles) with perimeter 36. They are (9, 10, 17), (9, 12, 15), (10, 10, 16), (10, 13, 13) and their areas are 36, 54, 48, 60 respectively.

Mathematica

gettriples[k_] := Select[IntegerPartitions[k, {3}], IntegerQ@Sqrt[Total@#*#[[1]]*#[[2]]*#[[3]]] &]; Table[Length@gettriples[k], {k, 1, 107}] (* Frank M Jackson, Jan 21 2026 *)

Crossrefs

Sequence in context: A227311 A178515 A344874 * A289359 A356771 A053118

Adjacent sequences: A051514 A051515 A051516 * A051518 A051519 A051520

Keyword

nonn

Author

David W. Wilson

Status

approved