A308068 Number of integer-sided triangles with perimeter n whose longest side length is even.

0, 0, 0, 0, 1, 1, 0, 0, 2, 2, 1, 1, 3, 3, 2, 2, 5, 5, 3, 3, 7, 7, 5, 5, 9, 9, 7, 7, 12, 12, 9, 9, 15, 15, 12, 12, 18, 18, 15, 15, 22, 22, 18, 18, 26, 26, 22, 22, 30, 30, 26, 26, 35, 35, 30, 30, 40, 40, 35, 35, 45, 45, 40, 40, 51, 51, 45, 45, 57, 57, 51, 51

Offset

1,9

Links

Table of n, a(n) for n=1..72.

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))) * ((n-i-k-1) mod 2).

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

G.f.: x^5 / ((1 - x)^3*(1 + x)^2*(1 - x + x^2)*(1 + x^2)^2*(1 + x + x^2)).

a(n) = a(n-1) - a(n-2) + a(n-3) + a(n-4) - a(n-5) + 2*a(n-6) - 2*a(n-7) + a(n-8) - a(n-9) - a(n-10) + a(n-11) - a(n-12) + a(n-13) for n>13.

(End)

Mathematica

Table[Sum[Sum[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

Sequence in context: A295515 A348890 A110659 * A100522 A258140 A140408

Adjacent sequences: A308065 A308066 A308067 * A308069 A308070 A308071

Keyword

nonn

Author

Wesley Ivan Hurt, May 10 2019

Status

approved