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

2,2

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).

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

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 *)

Crossrefs

Cf. A243207, A243212, A001399, A227327, A243209, A243210.

Sequence in context: A196741 A196899 A006411 * A373499 A129549 A171673

Adjacent sequences: A243205 A243206 A243207 * A243209 A243210 A243211

Keyword

nonn,easy

Author

Heinrich Ludwig, Jun 01 2014

Status

approved