|
%I #13 Aug 24 2020 12:36:16
%S 0,0,0,0,0,1,0,0,0,1,0,1,0,2,0,1,0,3,0,2,0,4,0,3,0,5,0,4,0,7,0,5,0,8,
%T 0,7,0,10,0,8,0,12,0,10,0,14,0,12,0,16,0,14,0,19,0,16,0,21,0,19,0,24,
%U 0,21,0,27,0,24,0,30,0,27,0,33,0,30,0,37,0
%N Number of triangles with perimeter n whose side lengths are even.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Integer_triangle">Integer Triangle</a>
%F a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} sign(floor((i+k)/(n-i-k+1))) * ((i-1) mod 2) * ((k-1) mod 2) * ((n-i-k-1) mod 2).
%F Conjectures from _Colin Barker_, May 11 2019: (Start)
%F G.f.: x^6 / ((1 - x)^3*(1 + x)^3*(1 - x + x^2)*(1 + x^2)^2*(1 + x + x^2)*(1 + x^4)).
%F a(n) = a(n-4) + a(n-6) + a(n-8) - a(n-10) - a(n-12) - a(n-14) + a(n-18) for n>18.
%F (End)
%t Table[Sum[Sum[Mod[i - 1, 2] Mod[k - 1, 2] Mod[n - i - k - 1, 2]*Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}]
%Y Cf. A308065.
%K nonn
%O 1,14
%A _Wesley Ivan Hurt_, May 10 2019
|
