A231653 Number of non-equivalent ways to choose 4 points in an equilateral triangle grid of side n.

0, 0, 4, 41, 244, 1029, 3485, 9926, 25030, 57126, 120570, 238330, 446344, 797825, 1370684, 2274259, 3660612, 5734776, 8771181, 13127940, 19270240, 27789713, 39435814, 55142010, 76066910, 103627784, 139554142, 185929971, 245260890, 320527585, 415268815

Offset: 1

Author

Heinrich Ludwig, Nov 12 2013

Keywords

• nonn
• easy

Examples

			For n = 3 there are the following a(3) = 4 choices of 4 points (=X) (rotations and reflections ignored):
    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 (2,3,-5,-8,3,19,4,-24,-15,15,24,-4,-19,-3,8,5,-3,-2,1).

Crossrefs

Cf. A234249, A001399, A227327, A230723, A231654, A231655.

Formula

a(n) = (n^8 + 4*n^7 - 6*n^6 - 32*n^5 + 84*n^4 - 32*n^3 - 16*n^2 - 192*n + B + C)/2304

where

B = 84*n^3 - 234*n^2 + 168*n + 171 if n==1 (mod 2)

= 0 otherwise

and

C = 128*n^2 + 128*n - 256 if n==1 (mod 3)

= 0 otherwise

G.f.: -x^3*(x^14 +7*x^12 +26*x^11 +146*x^10 +432*x^9 +947*x^8 +1418*x^7 +1621*x^6 +1405*x^5 +932*x^4 +438*x^3 +150*x^2 +33*x +4) / ((x -1)^9*(x +1)^4*(x^2 +x +1)^3). - Colin Barker, Feb 15 2014