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A282998
Number of ways to place 6 points on a triangular grid of side n so that no two of them are adjacent.
8
0, 0, 1, 353, 12231, 153194, 1124820, 5893221, 24425212, 85152341, 259805430, 712840480, 1793423456, 4197531636, 9240962666, 19301854131, 38514786780, 73828909906, 136581190475, 244784427831, 426389859697, 723857976770, 1200460734396, 1948846090829, 3102524331336
OFFSET
3,4
COMMENTS
Rotations and reflections of placements are counted. If they are to be ignored, see A279446.
a(n) is the coefficient of x^6 in the independence polynomial of the (n-1)-triangular grid graph. - Eric W. Weisstein, Mar 27 2026
LINKS
Eric Weisstein's World of Mathematics, Independence Polynomial.
Eric Weisstein's World of Mathematics, Triangular Grid Graph.
Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).
FORMULA
a(n) = (n^12 + 6*n^11 - 195*n^10 - 670*n^9 + 17455*n^8 + 13426*n^7 - 836249*n^6 + 1252990*n^5 + 19599884*n^4 - 68542552*n^3 - 131400416*n^2 + 974223360*n - 1308856320)/46080 for n>=4.
G.f.: x^5*(1 + 340*x + 7720*x^2 + 21439*x^3 - 12927*x^4 - 27265*x^5 + 28385*x^6 - 6252*x^7 - 116*x^8 - 2365*x^9 + 1787*x^10 - 352*x^11) / (1 - x)^13. - Colin Barker, Feb 26 2017
a(n) = 13*a(n-1)-78*a(n-2)+286*a(n-3)-715*a(n-4)+1287*a(n-5)-1716*a(n-6)+1716*a(n-7)-1287*a(n-8)+715*a(n-9)-286*a(n-10)+78*a(n-11)-13*a(n-12)+a(n-13). - Eric W. Weisstein, Mar 27 2026
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
MAPLE
A282998:=n->(n^12 + 6*n^11 - 195*n^10 - 670*n^9 + 17455*n^8 + 13426*n^7 - 836249*n^6 + 1252990*n^5 + 19599884*n^4 - 68542552*n^3 - 131400416*n^2 + 974223360*n - 1308856320)/46080: 0, seq(A282998(n), n=4..30); # Wesley Ivan Hurt, Apr 10 2017
MATHEMATICA
Drop[CoefficientList[Series[(x^5 * (1 + 340 * x + 7720 * x^2 + 21439 * x^3 - 12927 * x^4 - 27265 * x^5 + 28385 * x^6 - 6252 * x^7 - 116 * x^8 - 2365 * x^9 + 1787 * x^10 - 352 * x^11) / (1 - x)^13 ), {x, 0, 27}], x], 3] (* Indranil Ghosh, Feb 26 2017, from the g.f. by Colin Barker *)
seq = Join[{0}, Table[(n^12 + 6 n^11 - 195 n^10 - 670 n^9 + 17455 n^8 + 13426 n^7 - 836249 n^6 + 1252990 n^5 + 19599884 n^4 - 68542552 n^3 - 131400416 n^2 + 974223360 n - 1308856320)/46080, {n, 4, 50}]] (* Eric W. Weisstein, Mar 27 2026 *)
Join[{0}, LinearRecurrence[{13, -78, 286, -715, 1287, -1716, 1716, -1287, 715, -286, 78, -13, 1}, {0, 1, 353, 12231, 153194, 1124820, 5893221, 24425212, 85152341, 259805430, 712840480, 1793423456, 4197531636}, 20]] (* Eric W. Weisstein, Mar 27 2026 *)
PROG
(PARI) concat(vector(2), Vec(x^5*(1 + 340*x + 7720*x^2 + 21439*x^3 - 12927*x^4 - 27265*x^5 + 28385*x^6 - 6252*x^7 - 116*x^8 - 2365*x^9 + 1787*x^10 - 352*x^11) / (1 - x)^13 + O(x^30))) \\ Colin Barker, Feb 26 2017
CROSSREFS
Cf. A279446, A239567, A239568 (2 points), A239569 (3 points), A239570 (4 points), A239571 (5 points).
Sequence in context: A134820 A262205 A126113 * A213470 A068684 A377219
KEYWORD
nonn,easy
AUTHOR
Heinrich Ludwig, Feb 26 2017
STATUS
approved