A308108 - OEIS

%I #13 May 03 2021 14:11:20

%S 0,0,0,0,0,0,0,0,4,0,5,5,12,6,20,14,31,23,43,35,66,48,83,73,113,91,

%T 145,123,183,157,223,197,281,239,330,300,399,351,471,423,552,498,636,

%U 582,745,669,842,782,966,882,1094,1010,1234,1142,1378,1286,1554,1434

%N Sum of the largest side lengths of all integer-sided scalene triangles with perimeter n.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Integer_triangle">Integer Triangle</a>

%F a(n) = Sum_{k=1..floor((n-1)/3)} Sum_{i=k+1..floor((n-k-1)/2)} sign(floor((i+k)/(n-i-k+1))) * (n-i-k).

%F Conjectures from _Colin Barker_, May 13 2019: (Start)

%F G.f.: x^9*(2 + x + x^2)^2 / ((1 - x)^4*(1 + x)^3*(1 + x^2)^2*(1 + x + x^2)^2).

%F a(n) = -a(n-1) + 2*a(n-3) + 4*a(n-4) + 2*a(n-5) - a(n-6) - 5*a(n-7) - 5*a(n-8) - a(n-9) + 2*a(n-10) + 4*a(n-11) + 2*a(n-12) - a(n-14) - a(n-15) for n>15.

%F (End)

%t Table[Sum[Sum[(n - i - k)*Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k + 1,

%t Floor[(n - k - 1)/2]}], {k, Floor[(n - 1)/3]}], {n, 100}]

%Y Cf. A307966.

%K nonn

%O 1,9

%A _Wesley Ivan Hurt_, May 13 2019