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
KEYWORD
nonn,easy
AUTHOR
Heinrich Ludwig, Mar 23 2014
STATUS
approved