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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 (list; graph; refs; listen; history; text; internal format)
OFFSET

3,3

COMMENTS

Rotations and reflections of placements are not counted. If they are to be counted see A239571.

LINKS

Heinrich Ludwig, Table of n, a(n) for n = 3..1000

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

Adjacent sequences:  A239572 A239573 A239574 * A239576 A239577 A239578

KEYWORD

nonn,easy

AUTHOR

Heinrich Ludwig, Mar 23 2014

STATUS

approved

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Last modified April 1 14:39 EDT 2020. Contains 333163 sequences. (Running on oeis4.)