|
%I #12 Jun 13 2015 00:55:02
%S 0,3,20,77,223,552,1196,2380,4388,7657,12710,20301,31297,46892,68426,
%T 97674,136596,187713,253770,338217,444773,578018,742852,945210,
%U 1191398,1488949,1845824,2271415,2775605,3369930,4066480,4879238,5822810,6913947,8170098,9611127,11257671
%N Number of inequivalent (mod D_3) ways to place 3 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="/A243208/b243208.txt">Table of n, a(n) for n = 2..1000</a>
%H <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (3,0,-7,3,6,0,-6,-3,7,0,-3,1).
%F a(n) = (n^6 + 3*n^5 - 3*n^4 - 2*n^3 - 48*n^2 + 48*n)/288 + IF(MOD(n, 2) = 1)*(3*n^2 - 9*n - 1)/32 + IF(MOD(n, 3) = 1)*2/9.
%F G.f.: x^3*(-3 - 11*x - 17*x^2 - 13*x^3 - 14*x^4 - x^5 - 2*x^6 + x^7) / ((-1+x)^7 * (1+x)^3 * (1+x+x^2)). - _Vaclav Kotesovec_, Jun 02 2014
%F a(n) = 3*a(n-1) - 7*a(n-3) + 3*a(n-4) + 6*a(n-5) - 6*a(n-7) - 3*a(n-8) + 7*a(n-9) - 3*a(n-11) + a(n-12). - _Vaclav Kotesovec_, Jun 02 2014
%t Drop[CoefficientList[Series[x^3*(-3 - 11*x - 17*x^2 - 13*x^3 - 14*x^4 - x^5 - 2*x^6 + x^7) / ((-1+x)^7 * (1+x)^3 * (1+x+x^2)), {x, 0, 50}], x],2] (* _Vaclav Kotesovec_, Jun 02 2014 *)
%Y Cf. A243207, A243212, A001399, A227327, A243209, A243210.
%K nonn,easy
%O 2,2
%A _Heinrich Ludwig_, Jun 01 2014
|
