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

1,32

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.

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

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 *)

Crossrefs

Cf. A024153, A070139.

Sequence in context: A336644 A135416 A134309 * A236799 A208274 A208604

Adjacent sequences: A051513 A051514 A051515 * A051517 A051518 A051519

Keyword

nonn

Author

David W. Wilson

Status

approved