|
| |
|
|
A239574
|
|
Number of non-equivalent (mod D_3) ways to place 4 indistinguishable points on a triangular grid of side n so that no two of them are adjacent.
|
|
6
|
|
|
|
0, 1, 24, 200, 1053, 3932, 11988, 31298, 73046, 155880, 310046, 581414, 1038634, 1779531, 2942114, 4714412, 7350595, 11184786, 16654116, 24317554, 34886940, 49252544, 68523846, 94062350, 127534794, 170954603, 226748678, 297809946, 387580007, 500113190, 640178710
(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 A239570.
|
|
|
LINKS
|
Index entries for linear recurrences with constant coefficients, signature (2,3,-5,-8,3,19,4,-24,-15,15,24,-4,-19,-3,8,5,-3,-2,1)
|
|
|
FORMULA
|
a(n) = (n^8 +4*n^7 -78*n^6 -104*n^5 +2556*n^4 -3152*n^3 -27280*n^2 +89664*n -78336)/2304 +IF(n == 1 mod 2)*(28*n^3 -54*n^2 -160*n +129)/768 +IF(n == 1 mod 3)*(n^2 +n -14)/18.
G.f.: x^4*(-1 - 22*x - 149*x^2 - 586*x^3 - 1354*x^4 - 2154*x^5 - 2300*x^6 - 1510*x^7 - 259*x^8 + 470*x^9 + 443*x^10 + 70*x^11 - 130*x^12 - 94*x^13 - 10*x^14 + 18*x^15 + 8*x^16) / ((-1+x)^9 * (1+x)^4 * (1+x+x^2)^3). - Vaclav Kotesovec, Mar 29 2014
|
|
|
EXAMPLE
|
There is a(4) = 1 way to place 4 points on a triangular grid of side n = 4:
X
. .
. X .
X . . X
|
|
|
MATHEMATICA
|
Drop[CoefficientList[Series[x^4*(-1 - 22*x - 149*x^2 - 586*x^3 - 1354*x^4 - 2154*x^5 - 2300*x^6 - 1510*x^7 - 259*x^8 + 470*x^9 + 443*x^10 + 70*x^11 - 130*x^12 - 94*x^13 - 10*x^14 + 18*x^15 + 8*x^16) / ((-1+x)^9 * (1+x)^4 * (1+x+x^2)^3), {x, 0, 20}], x], 3] (* Vaclav Kotesovec, Mar 29 2014 *)
Table[(n^8+4*n^7-78*n^6-104*n^5+2556*n^4-3152*n^3-27280*n^2+89664*n-78336)/2304 + If[Mod[n, 2]==1, (28*n^3-54*n^2-160*n+129)/768, 0] + If[Mod[n, 3]==1, (n^2+n-14)/18, 0], {n, 3, 20}] (* Vaclav Kotesovec after Heinrich Ludwig, Mar 29 2014 *)
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,easy
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
