A308091 Sum of the areas of the integer-sided triangles with perimeter n and integer area.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 12, 0, 12, 0, 0, 0, 0, 0, 24, 0, 0, 0, 0, 0, 30, 0, 72, 0, 0, 0, 198, 0, 0, 0, 60, 0, 126, 0, 66, 0, 0, 0, 288, 0, 180, 0, 0, 0, 360, 0, 84, 0, 0, 0, 330, 0, 0, 0, 648, 0, 132, 0, 204, 0, 420, 0, 876, 0, 0, 0, 114

Offset: 1

Author

Wesley Ivan Hurt, May 12 2019

Keywords

• nonn

Links

• Wikipedia, Integer Triangle

Crossrefs

Cf. A051516.

Programs

Mathematica

Table[Sum[Sum[Sqrt[(n/2) (n/2 - i) (n/2 - k) (i + k - n/2)] (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}]

Formula

a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} m * (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)).