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A279446
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Number of non-equivalent (mod D_3) ways to place 6 indistinguishable points on a triangular grid of side n so that no two of them are adjacent.
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5
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0, 0, 1, 66, 2096, 25676, 187984, 983172, 4073312, 14196011, 43309138, 118818916, 298926225, 699619679, 1540212590, 3217045155, 6419240369, 12304959047, 22763742133, 40797668697, 71065355815, 120643462032, 200077436639, 324808463585, 517088445952, 808515893580
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OFFSET
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3,4
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COMMENTS
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Rotations and reflections of placements are not counted. For numbers if they are to be counted see A282998.
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (4,0,-17,8,36,-7,-68,-18,113,52,-126,-92,92,126,-52,-113,18,68,7,-36,-8,17,0,-4,1).
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FORMULA
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a(n) = (n^12 + 6*n^11 - 195*n^10 - 670*n^9 + 17455*n^8 + 13426*n^7 - 835256*n^6 + 1246240*n^5 + 19563664*n^4 - 68181792*n^3 - 131524224*n^2 + 969500160*n - 1298903040)/276480 + IF(MOD(n, 2) = 1, 162*n^5 - 715*n^4 - 4480*n^3 + 21955*n^2 + 1108*n - 41685)/30720 + IF(MOD(n, 3) = 1, n^2 + n - 25)/27 for n>=4.
G.f.: x^5*(1 + 62*x + 1832*x^2 + 17309*x^3 + 86394*x^4 + 266304*x^5 + 557979*x^6 + 818157*x^7 + 829988*x^8 + 519203*x^9 + 94134*x^10 - 150065*x^11 - 123434*x^12 + 7445*x^13 + 64052*x^14 + 29943*x^15 - 11247*x^16 - 15803*x^17 - 3012*x^18 + 3100*x^19 + 1722*x^20 - 15*x^21 - 233*x^22 - 56*x^23) / ((1 - x)^13*(1 + x)^6*(1 + x + x^2)^3). - Colin Barker, Feb 26 2017
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EXAMPLE
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There is a(5) = 1 way to place 6 points on a triangular grid of side n = 5:
X
. .
X . X
. . . .
X . X . X
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MATHEMATICA
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Table[Boole[n > 4] ((n^12 + 6 n^11 - 195 n^10 - 670 n^9 + 17455 n^8 + 13426 n^7 - 835256 n^6 + 1246240 n^5 + 19563664 n^4 - 68181792 n^3 - 131524224 n^2 + 969500160 n - 1298903040)/276480 + Boole[OddQ@ n] (162 n^5 - 715 n^4 - 4480 n^3 + 21955 n^2 + 1108 n - 41685)/30720 + Boole[Mod[n, 3] == 1] (n^2 + n - 25)/27), {n, 3, 28}] (* Michael De Vlieger, Feb 26 2017 *)
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PROG
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(PARI) concat(vector(2), Vec(x^5*(1 + 62*x + 1832*x^2 + 17309*x^3 + 86394*x^4 + 266304*x^5 + 557979*x^6 + 818157*x^7 + 829988*x^8 + 519203*x^9 + 94134*x^10 - 150065*x^11 - 123434*x^12 + 7445*x^13 + 64052*x^14 + 29943*x^15 - 11247*x^16 - 15803*x^17 - 3012*x^18 + 3100*x^19 + 1722*x^20 - 15*x^21 - 233*x^22 - 56*x^23) / ((1 - x)^13*(1 + x)^6*(1 + x + x^2)^3) + O(x^30))) \\ Colin Barker, Feb 26 2017
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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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