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A243209
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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.
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5
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1, 25, 186, 881, 3146, 9264, 23810, 55058, 117205, 233135, 438544, 786541, 1354696, 2252202, 3630684, 5694984, 8718963, 13060515, 19184110, 27681103, 39300096, 54974216, 75861038, 103377456, 139251749, 185567453, 244828780, 320015885, 414665890, 532940080
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OFFSET
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3,2
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (3,0,-6,0,6,8,-12,-9,13,6,-6,-13,9,12,-8,-6,0,6,0,-3,1).
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FORMULA
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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).
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
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
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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