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

1,3

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)

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

Cf. A001399, A014409, A082966, A227327.

Sequence in context: A133714 A164271 A233698 * A220275 A055585 A099625

Adjacent sequences: A230720 A230721 A230722 * A230724 A230725 A230726

Keyword

nonn,easy

Author

Heinrich Ludwig, Oct 28 2013

Status

approved