A243213 Number of ways to place 4 points on a triangular grid of side length n so that no three of them are vertices of an equilateral triangle with sides parallel to the grid.

3, 128, 1062, 5160, 18591, 55113, 142005, 329045, 701160, 1395975, 2626953, 4713723, 8120322, 13503350, 21770766, 34153758, 52292385, 78337890, 115072320, 166048850, 235753353, 329791143, 455099307, 620189115, 835418766, 1113301553, 1468849515, 1919958285

Offset

3,1

Links

Heinrich Ludwig, Table of n, a(n) for n = 3..1000

Index entries for linear recurrences with constant coefficients, signature (6,-12,2,27,-36,0,36,-27,-2,12,-6,1)

Formula

a(n) = (n^8 + 4*n^7 - 6*n^6 - 80*n^5 - 15*n^4 + 532*n^3 - 244*n^2 - 432*n)/384 + IF(MOD(n, 2) = 1)*(-n^2 - n + 12)/16.

G.f.: x^3*(7*x^7-33*x^6-15*x^5-38*x^4-318*x^3-330*x^2-110*x-3) / ((x-1)^9*(x+1)^3). - Colin Barker, Jun 09 2014

Example

There are exactly a(3) = 3 ways to place 4 points (x) on a 3X3X3 grid, no three of them being vertices of an equilateral triangle:
      .            x            x
     x x          . x          x .
    x . x        x x .        . x x

Prog

(PARI) Vec(x^3*(7*x^7-33*x^6-15*x^5-38*x^4-318*x^3-330*x^2-110*x-3)/((x-1)^9*(x+1)^3) + O(x^100)) \\ Colin Barker, Jun 09 2014

Crossrefs

Cf. A243211, A243209, A000217, A050534, A243212, A243214.

Sequence in context: A134711 A163850 A258671 * A264582 A108713 A319222

Adjacent sequences: A243210 A243211 A243212 * A243214 A243215 A243216

Keyword

nonn,easy

Author

Heinrich Ludwig, Jun 09 2014

Status

approved