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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.
6

%I #21 Oct 10 2017 22:35:12

%S 0,1,24,200,1053,3932,11988,31298,73046,155880,310046,581414,1038634,

%T 1779531,2942114,4714412,7350595,11184786,16654116,24317554,34886940,

%U 49252544,68523846,94062350,127534794,170954603,226748678,297809946,387580007,500113190,640178710

%N 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.

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

%H Heinrich Ludwig, <a href="/A239574/b239574.txt">Table of n, a(n) for n = 3..1000</a>

%H <a href="/index/Rec#order_19">Index entries for linear recurrences with constant coefficients</a>, signature (2,3,-5,-8,3,19,4,-24,-15,15,24,-4,-19,-3,8,5,-3,-2,1)

%F 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.

%F 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

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

%e X

%e . .

%e . X .

%e X . . X

%t 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 *)

%t 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 *)

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

%K nonn,easy

%O 3,3

%A _Heinrich Ludwig_, Mar 23 2014