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A239573 Number of non-equivalent (mod D_3) ways to place 3 indistinguishable points on a triangular grid of side n so that no two of them are adjacent. 6
0, 1, 6, 32, 113, 329, 790, 1702, 3320, 6057, 10400, 17074, 26903, 41047, 60796, 87886, 124220, 172275, 234732, 314992, 416703, 544391, 702878, 898040, 1136098, 1424521, 1771178, 2185392, 2676947, 3257305, 3938450, 4734286, 5659306, 6730177, 7964228, 9381234 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,3

COMMENTS

Rotations and reflections of placements are not counted. If they are to be counted see A239569.

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 - 39*n^4 + 10*n^3 + 456*n^2 - 1008*n + 576)/288 + IF(MOD(n, 2) = 1)*(3*n^2 - 5*n - 5)/32 + IF(MOD(n, 3) = 1)*2/9.

G.f.: -x^3*(2*x^9 +x^8 -8*x^7 -9*x^6 +3*x^5 +29*x^4 +24*x^3 +14*x^2 +3*x +1) / ((x -1)^7*(x +1)^3*(x^2 +x +1)). - Colin Barker, Mar 23 2014

EXAMPLE

There are a(4) = 6 non-equivalent ways to place 3 points on a triangular grid of side 4:

    .           X           X           X           X           X

   . X         . .         . .         . .         . .         . .

  X . .       X . X       X . .       X . .       . X .       . . .

. . X .     . . . .     . . X .     . . . X     . . . X     X . . X

PROG

(PARI) concat(0, Vec(-x^3*(2*x^9 +x^8 -8*x^7 -9*x^6 +3*x^5 +29*x^4 +24*x^3 +14*x^2 +3*x +1)/((x -1)^7*(x +1)^3*(x^2 +x +1)) + O(x^100))) \\ Colin Barker, Mar 23 2014

CROSSREFS

Cf. A239572, A239569, A032091 (2 points), A239574 (4 points), A239575 (5 points), A279446 (6 points).

Sequence in context: A102359 A000397 A200765 * A130410 A202807 A203324

Adjacent sequences:  A239570 A239571 A239572 * A239574 A239575 A239576

KEYWORD

nonn,easy

AUTHOR

Heinrich Ludwig, Mar 23 2014

STATUS

approved

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Last modified April 18 10:56 EDT 2021. Contains 343087 sequences. (Running on oeis4.)