A308303 Number of integer-sided triangles with perimeter n and at least one even side length.

0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 3, 4, 4, 4, 5, 6, 7, 6, 8, 9, 10, 9, 12, 12, 14, 12, 16, 16, 19, 16, 21, 20, 24, 20, 27, 25, 30, 25, 33, 30, 37, 30, 40, 36, 44, 36, 48, 42, 52, 42, 56, 49, 61, 49, 65, 56, 70, 56, 75, 64, 80, 64, 85, 72, 91, 72, 96, 81, 102

Offset

1,9

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

a(n) = A005044(n) - A308066(n+3).

G.f.: x^5*(1 + x + x^2 + x^3 + x^4 + x^5 + x^7)/((1 - x)^3*(1 + x)^3*(1 - x + x^2)*(1 + x^2)^2*(1 + x + x^2)*(1 + x^4)). - Andrew Howroyd, Nov 06 2025

a(n) = a(n-4) + a(n-6) + a(n-8) - a(n-10) - a(n-12) - a(n-14) + a(n-18). - Wesley Ivan Hurt, Nov 29 2025

Mathematica

Table[Sum[Sum[(1 - Mod[i, 2] Mod[k, 2] Mod[n - i - k, 2])*Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}]

Crossrefs

Cf. A005044, A308066.

Sequence in context: A265409 A126257 A025773 * A285763 A294621 A029077

Adjacent sequences: A308300 A308301 A308302 * A308304 A308305 A308306

Keyword

nonn

Author

Wesley Ivan Hurt, May 19 2019

Status

approved