Equilateral triangles in lattice cube

[Joshua Zucker]

Hi folks,
I just submitted the following to OEIS.
My algorithm for calculating the terms is complete brute force and
will take a while to run to get even 1 more term ...
can anyone help me with more terms, or with an idea for a faster algorithm?

Thanks,
--Joshua Zucker

%S A000001 8, 80, 368, 1264, 3448, 7792
%N A000001 Equilateral triangles in the lattice cube of side length n.
%D A000001 Inspired by problem 25 on the 2005 AMC-12A mathematics
competition, which asked for a(2).
%e A000001 a(1) = 8 because in the unit cube, equilateral triangles
are formed by cutting off any one of the 8 corners.
a(2) = 80 because there are 8 unit cubes with 8 each, 8 larger
triangles (analogous to the 8 in the unit cube, but twice as big), and
also 8 triangles of side length sqrt(6).
%O A000001 1
%K A000001 ,more,nonn,
%A A000001 Joshua Zucker (joshua.zucker at stanfordalumni.org), Feb 04 2005