A243143 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 of any orientation.
1, 22, 170, 816, 2947, 8765, 22703, 52823, 113042, 225817, 426299, 766905, 1324282, 2206478, 3563770, 5599258, 8584775, 12875840, 18934040, 27347390, 38860741, 54402707, 75125825, 102441321, 138070912, 184090795, 242997153, 317760863, 411908932, 529591532, 675681764
Offset: 3
Links
- Heinrich Ludwig, Table of n, a(n) for n = 3..1000
- Index entries for linear recurrences with constant coefficients, signature (5,-7,-5,23,-19,-7,27,-27,7,19,-23,5,7,-5,1).
Programs
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Mathematica
Drop[CoefficientList[Series[-x^3*(3*x^10 - 10*x^9 + 19*x^8 - 13*x^7 + 102*x^6 + 105*x^5 + 144*x^4 + 125*x^3 + 67*x^2 + 17*x + 1) / ((x-1)^9*(x+1)^4*(x^2+1)), {x, 0, 40}], x],3] (* Vaclav Kotesovec, May 31 2014 after Colin Barker *)
Formula
a(n) = (n^8 + 4*n^7 - 14*n^6 - 56*n^5 + 136*n^4 - 104*n^3 + 552*n^2 - 672*n)/2304 + IF(MOD(n, 2) = 1)*(28*n^3 - 198*n^2 + 296*n + 21)/768 + IF(MOD(n-1, 4) <= 1)*(-1/8).
G.f.: -x^3*(3*x^10 -10*x^9 +19*x^8 -13*x^7 +102*x^6 +105*x^5 +144*x^4 +125*x^3 +67*x^2 +17*x +1) / ((x -1)^9*(x +1)^4*(x^2 +1)). - Colin Barker, May 30 2014