A239574 Number of non-equivalent (mod D_3) ways to place 4 indistinguishable points on a triangular grid of side n so that no two of them are adjacent.

0, 1, 24, 200, 1053, 3932, 11988, 31298, 73046, 155880, 310046, 581414, 1038634, 1779531, 2942114, 4714412, 7350595, 11184786, 16654116, 24317554, 34886940, 49252544, 68523846, 94062350, 127534794, 170954603, 226748678, 297809946, 387580007, 500113190, 640178710

Offset

3,3

Comments

Rotations and reflections of placements are not counted. If they are to be counted see A239570.

Links

Heinrich Ludwig, Table of n, a(n) for n = 3..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 -78*n^6 -104*n^5 +2556*n^4 -3152*n^3 -27280*n^2 +89664*n -78336)/2304 +IF(n == 1 mod 2)*(28*n^3 -54*n^2 -160*n +129)/768 +IF(n == 1 mod 3)*(n^2 +n -14)/18.

G.f.: x^4*(-1 - 22*x - 149*x^2 - 586*x^3 - 1354*x^4 - 2154*x^5 - 2300*x^6 - 1510*x^7 - 259*x^8 + 470*x^9 + 443*x^10 + 70*x^11 - 130*x^12 - 94*x^13 - 10*x^14 + 18*x^15 + 8*x^16) / ((-1+x)^9 * (1+x)^4 * (1+x+x^2)^3). - Vaclav Kotesovec, Mar 29 2014

Example

There is a(4) = 1 way to place 4 points on a triangular grid of side n = 4:
      X
     . .
    . X .
   X . . X

Mathematica

Drop[CoefficientList[Series[x^4*(-1 - 22*x - 149*x^2 - 586*x^3 - 1354*x^4 - 2154*x^5 - 2300*x^6 - 1510*x^7 - 259*x^8 + 470*x^9 + 443*x^10 + 70*x^11 - 130*x^12 - 94*x^13 - 10*x^14 + 18*x^15 + 8*x^16) / ((-1+x)^9 * (1+x)^4 * (1+x+x^2)^3), {x, 0, 20}], x], 3] (* Vaclav Kotesovec, Mar 29 2014 *)
Table[(n^8+4*n^7-78*n^6-104*n^5+2556*n^4-3152*n^3-27280*n^2+89664*n-78336)/2304 + If[Mod[n, 2]==1, (28*n^3-54*n^2-160*n+129)/768, 0] + If[Mod[n, 3]==1, (n^2+n-14)/18, 0], {n, 3, 20}] (* Vaclav Kotesovec after Heinrich Ludwig, Mar 29 2014 *)

Crossrefs

Cf. A239572, A239570, A032091 (2 points), A239573 (3 points), A239575 (5 points), A279446 (6 points).

Sequence in context: A048355 A132458 A055857 * A223748 A269691 A282644

Adjacent sequences: A239571 A239572 A239573 * A239575 A239576 A239577

Keyword

nonn,easy

Author

Heinrich Ludwig, Mar 23 2014

Status

approved