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

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, 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, 0, 21, 0, 27, 0, 24, 0, 30, 0, 27, 0, 33, 0, 30, 0, 37, 0

Offset

1,14

Links

Table of n, a(n) for n=1..79.

Wikipedia, Integer Triangle

Formula

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).

Conjectures from Colin Barker, May 11 2019: (Start)

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)).

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.

(End)

Mathematica

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}]

Crossrefs

Cf. A308065.

Sequence in context: A108044 A104477 A224928 * A052173 A177825 A175790

Adjacent sequences: A308063 A308064 A308065 * A308067 A308068 A308069

Keyword

nonn

Author

Wesley Ivan Hurt, May 10 2019

Status

approved