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

1,3

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).

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

Example

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

Crossrefs

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

Sequence in context: A068169 A069631 A069616 * A057419 A089664 A121671

Adjacent sequences: A231650 A231651 A231652 * A231654 A231655 A231656

Keyword

nonn,easy

Author

Heinrich Ludwig, Nov 12 2013

Status

approved