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 A308066 Number of triangles with perimeter n whose side lengths are even. 0

%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

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Last modified August 14 13:17 EDT 2024. Contains 375165 sequences. (Running on oeis4.)