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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 (list; graph; refs; listen; history; text; internal format)
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 .

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

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Last modified May 25 19:13 EDT 2019. Contains 323576 sequences. (Running on oeis4.)