A051516 Number of triangles with perimeter n having integer sides and area.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 4, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 0, 3, 0, 2, 0, 0, 0, 4, 0, 1, 0, 0, 0, 3, 0, 0, 0, 5, 0, 1, 0, 1, 0, 2, 0, 5, 0, 0, 0, 1, 0, 1, 0, 4, 0, 0, 0, 8, 0, 0, 0, 1, 0, 5, 0, 0, 0, 0, 0, 5, 0, 6, 0, 5, 0, 0, 0, 2, 0, 0, 0, 12, 0, 1, 0

Offset: 1

Author

David W. Wilson

Keywords

• nonn

Comments

No such triangles with odd perimeter exist.

Links

• Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 1000 terms from Seiichi Manyama)
• Eric Weisstein's World of Mathematics, Heronian Triangle.

Crossrefs

Cf. A024153, A070139.

Programs

Mathematica

Table[Sum[Sum[(1 - Ceiling[Sqrt[(n/2) (n/2 - i) (n/2 - k) (i + k - n/2)]] + Floor[Sqrt[(n/2) (n/2 - i) (n/2 - k) (i + k - n/2)]])*Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}] (* Wesley Ivan Hurt, May 11 2019 *)

Formula

a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} (1 - ceiling(m) + floor(m)) * sign(floor((i+k)/(n-i-k+1))), where m = sqrt((n/2)*(n/2-i)*(n/2-k)*(i+k-n/2)). - Wesley Ivan Hurt, May 11 2019