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URL: https://oeis.org/A308066

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A308066
Number of triangles with perimeter n whose side lengths are even.
3
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
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
KEYWORD
nonn
AUTHOR
Wesley Ivan Hurt, May 10 2019
STATUS
approved