login
A279446
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.
5
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
OFFSET
3,4
COMMENTS
Rotations and reflections of placements are not counted. For numbers if they are to be counted see A282998.
LINKS
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).
FORMULA
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
EXAMPLE
There is a(5) = 1 way to place 6 points on a triangular grid of side n = 5:
X
. .
X . X
. . . .
X . X . X
MATHEMATICA
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 *)
PROG
(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
CROSSREFS
Cf. A282998, A239572, A032091 (2 points), A239573 (3 points), A239574 (4 points), A239575 (5 points).
Sequence in context: A270847 A140925 A269779 * A271797 A017782 A035728
KEYWORD
nonn,easy
AUTHOR
Heinrich Ludwig, Feb 26 2017
STATUS
approved