A243208 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.
0, 3, 20, 77, 223, 552, 1196, 2380, 4388, 7657, 12710, 20301, 31297, 46892, 68426, 97674, 136596, 187713, 253770, 338217, 444773, 578018, 742852, 945210, 1191398, 1488949, 1845824, 2271415, 2775605, 3369930, 4066480, 4879238, 5822810, 6913947, 8170098, 9611127, 11257671
Offset: 2
Links
- Heinrich Ludwig, Table of n, a(n) for n = 2..1000
- Index entries for linear recurrences with constant coefficients, signature (3,0,-7,3,6,0,-6,-3,7,0,-3,1).
Programs
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Mathematica
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 *)
Formula
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.
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
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