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%I #14 Jun 13 2015 00:55:03
%S 1,25,186,881,3146,9264,23810,55058,117205,233135,438544,786541,
%T 1354696,2252202,3630684,5694984,8718963,13060515,19184110,27681103,
%U 39300096,54974216,75861038,103377456,139251749,185567453,244828780,320015885,414665890,532940080
%N Number of inequivalent (mod D_3) ways to place 4 points on a triangular grid of side n so that they are not vertices of an equilateral triangle with sides parallel to the grid.
%H Heinrich Ludwig, <a href="/A243209/b243209.txt">Table of n, a(n) for n = 3..1000</a>
%H <a href="/index/Rec#order_21">Index entries for linear recurrences with constant coefficients</a>, signature (3,0,-6,0,6,8,-12,-9,13,6,-6,-13,9,12,-8,-6,0,6,0,-3,1).
%F a(n) = (n^8 + 4*n^7 - 6*n^6 - 80*n^5 + 60*n^4 + 208*n^3 + 464*n^2 - 1152*n)/2304 + IF(MOD(n, 2) = 1)*(28*n^3 - 206*n^2 + 312*n + 33)/768 + IF(MOD(n, 3) = 1)*(n^2 - 2*n + 4)/18 + IF(MOD(n, 6) = 1)*(- 1/6).
%F G.f.: -x^3*(1 + 22*x + 111*x^2 + 329*x^3 + 653*x^4 + 936*x^5 + 1146*x^6 + 1200*x^7 + 1150*x^8 + 900*x^9 + 650*x^10 + 286*x^11 + 131*x^12 + 28*x^13 + 19*x^14 - 5*x^15 + 3*x^16) / ((-1+x)^9 * (1+x)^4 * (1-x+x^2) * (1+x+x^2)^3). - _Vaclav Kotesovec_, Jun 02 2014
%F a(n) = 3*a(n-1) - 6*a(n-3) + 6*a(n-5) + 8*a(n-6) - 12*a(n-7) - 9*a(n-8) + 13*a(n-9) + 6*a(n-10) - 6*a(n-11) - 13*a(n-12) + 9*a(n-13) + 12*a(n-14) - 8*a(n-15) - 6*a(n-16) + 6*a(n-18) - 3*a(n-20) + a(n-21). - _Vaclav Kotesovec_, Jun 02 2014
%t Drop[CoefficientList[Series[-x^3*(1 + 22*x + 111*x^2 + 329*x^3 + 653*x^4 + 936*x^5 + 1146*x^6 + 1200*x^7 + 1150*x^8 + 900*x^9 + 650*x^10 + 286*x^11 + 131*x^12 + 28*x^13 + 19*x^14 - 5*x^15 + 3*x^16) / ((-1+x)^9 * (1+x)^4 * (1-x+x^2) * (1+x+x^2)^3), {x, 0, 40}], x],3] (* _Vaclav Kotesovec_, Jun 02 2014 *)
%Y Cf. A243207, A243213, A001399, A227327, A243208, A243210.
%K nonn,easy
%O 3,2
%A _Heinrich Ludwig_, Jun 01 2014
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