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A239575
Number of non-equivalent (mod D_3) ways to place 5 indistinguishable points on a triangular grid of side n so that no two of them are adjacent.
6
0, 0, 7, 176, 1976, 12565, 57275, 207018, 634166, 1711262, 4181915, 9428657, 19892816, 39684027, 75473209, 137721045, 242391212, 413215132, 684733527, 1106194950, 1746637600, 2701244609, 4099429895, 6114748948, 8977257362, 12988406970, 18539308619, 26132434991
OFFSET
3,3
COMMENTS
Rotations and reflections of placements are not counted. If they are to be counted see A239571.
LINKS
Index entries for linear recurrences with constant coefficients, signature (6,-10,-10,50,-34,-66,110,0,-110,66,34,-50,10,10,-6,1)
FORMULA
a(n) = (n^10 + 5*n^9 - 130*n^8 - 310*n^7 + 7465*n^6 - 1336*n^5 - 202980*n^4 + 464160*n^3 + 1783424*n^2 - 8360064*n + 9192960)/23040 + IF(MOD(n,2) = 1)*(25*n^4 - 94*n^3 - 418*n^2 + 2053*n - 1779)/1536.
G.f.: x^2*(-19 - (19 - 114*x + 190*x^2 + 197*x^3 - 816*x^4 + 1636*x^5 + 3793*x^6 + 965*x^7 + 216*x^8 + 194*x^9 - 2278*x^10 + 53*x^11 + 1547*x^12 - 336*x^13 - 351*x^14 + 125*x^15) / ((-1+x)^11 * (1+x)^5)). - Vaclav Kotesovec, Mar 31 2014
EXAMPLE
There are a(5) = 7 non-equivalent ways to place 5 points (x) on a triangular grid of side 5. These are:
x x . x
. . . . . . . .
x . x x . x x . x . x .
. . . . . . . . . . . . . . . .
x . . . x . x . x . x . x . x x . x . x
.
x x x
. . . . . .
. x . . x . x . x
x . . x x . . . . . . .
. . x . . . . x . x x . . x .
MATHEMATICA
Table[(n^10 + 5*n^9 - 130*n^8 - 310*n^7 + 7465*n^6 - 1336*n^5 - 202980*n^4 + 464160*n^3 + 1783424*n^2 - 8360064*n + 9192960)/23040 + (1-(-1)^n)/2*(25*n^4 - 94*n^3 - 418*n^2 + 2053*n - 1779)/1536, {n, 3, 20}] (* Vaclav Kotesovec after Heinrich Ludwig, Mar 31 2014 *)
Drop[CoefficientList[Series[x^2*(-19 - (19 - 114*x + 190*x^2 + 197*x^3 - 816*x^4 + 1636*x^5 + 3793*x^6 + 965*x^7 + 216*x^8 + 194*x^9 - 2278*x^10 + 53*x^11 + 1547*x^12 - 336*x^13 - 351*x^14 + 125*x^15) / ((-1+x)^11*(1+x)^5)), {x, 0, 20}], x], 3] (* Vaclav Kotesovec, Mar 31 2014 *)
CROSSREFS
Cf. A239572, A239571, A032091 (2 points), A239573 (3 points), A239574 (4 points), 279446 (6 points).
Sequence in context: A195887 A162082 A195517 * A152930 A027489 A098433
KEYWORD
nonn,easy
AUTHOR
Heinrich Ludwig, Mar 23 2014
STATUS
approved