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

Comments

a(n+3) is also the number of triangles with perimeter n whose side lengths are odd. - Andrew Howroyd, Nov 06 2025

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Wikipedia, Integer Triangle

Index entries for linear recurrences with constant coefficients, signature (0,0,0,1,0,1,0,1,0,-1,0,-1,0,-1,0,0,0,1).

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)

From Andrew Howroyd, Nov 06 2025: (Start)

The above conjectures are correct.

a(n) = A005044(n/2) for even n; a(n) = 0 for odd n. (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. A005044.

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