A308097 Take the integer-sided triangles with perimeter n and integer area. Then a(n) is the sum of the areas of all the triangles and the squares on their sides.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 56, 0, 0, 0, 98, 0, 126, 0, 0, 0, 0, 0, 224, 0, 0, 0, 0, 0, 368, 0, 826, 0, 0, 0, 2012, 0, 0, 0, 638, 0, 1390, 0, 756, 0, 0, 0, 2692, 0, 1928, 0, 0, 0, 4764, 0, 1334, 0, 0, 0, 4434, 0, 0, 0, 8354, 0, 1778, 0, 1794, 0, 3800, 0

Offset

1,12

Links

Table of n, a(n) for n=1..71.

Wikipedia, Integer Triangle

Formula

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

Mathematica

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

Crossrefs

Cf. A308091.

Sequence in context: A201823 A115410 A250489 * A172533 A206786 A036197

Adjacent sequences: A308094 A308095 A308096 * A308098 A308099 A308100

Keyword

nonn

Author

Wesley Ivan Hurt, May 12 2019

Status

approved