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A230723
Number of non-equivalent ways to choose three points in an equilateral triangle grid of side n.
4
0, 1, 6, 25, 87, 238, 575, 1228, 2425, 4446, 7734, 12806, 20422, 31444, 47072, 68639, 97929, 136893, 188061, 254170, 338679, 445297, 578616, 743524, 945968, 1192243, 1489894, 1846869, 2272575, 2776880, 3371335, 4068016, 4880921, 5824640, 6915942, 8172258, 9613470
OFFSET
1,3
LINKS
FORMULA
a(n) = (n^6 + 3*n^5 - 3*n^4 + 10*n^3 + B + C)/288
where
B = 27*n^2 + 3*n - 9 if n odd
B = 48*n otherwise
and
C = -32 if n == 1 (mod 3)
C = 0 otherwise
G.f.: x^2*(1 + 3*x + 7*x^2 + 19*x^3 + 16*x^4 + 12*x^5 + x^6 + 2*x^7 - x^8)/((1-x^3) * (1-x^2)^3 * (1-x)^3). - Ralf Stephan, Nov 03 2013
EXAMPLE
for n = 3 there are the following a(3) = 6 choices of 3 points (=X) (rotations and reflections ignored):
X . . X . X
. . X X . . X X . X X .
X . X . X . X X X . . . X . X . X .
MATHEMATICA
LinearRecurrence[{3, 0, -7, 3, 6, 0, -6, -3, 7, 0, -3, 1}, {0, 1, 6, 25, 87, 238, 575, 1228, 2425, 4446, 7734, 12806}, 40] (* Harvey P. Dale, Oct 24 2020 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Heinrich Ludwig, Oct 28 2013
STATUS
approved