A230723 Number of non-equivalent ways to choose three points in an equilateral triangle grid of side n.
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
Examples
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 .
Links
- Heinrich Ludwig, Table of n, a(n) for n = 1..1000
- Index entries for linear recurrences with constant coefficients, signature (3,0,-7,3,6,0,-6,-3,7,0,-3,1)
Programs
-
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 *)
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