%I #25 Sep 08 2022 08:46:07

%S 0,0,27,999,11565,74811,342042,1239525,3799488,10259640,25076952,

%T 56552364,119324403,238062357,452774595,826245798,1454229216,

%U 2479147536,4108199481,6636929805,10479498849,16207085223,24596072424,36687908235,53862785520,77929575480

%N Number of ways to place 5 points on a triangular grid of side n so that no two of them are adjacent.

%C Rotations and reflections of placements are counted. If they are to be ignored see A239575.

%H Vincenzo Librandi, <a href="/A239571/b239571.txt">Table of n, a(n) for n = 3..1000</a>

%H <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1)

%F a(n) = (n -3) * (n -4) * (n^8 +12*n^7 -58*n^6 -860*n^5 +2141*n^4 +23728*n^3 -61316*n^2 -244928*n +770880)/3840.

%F G.f.: -3*x^5*(40*x^8-185*x^7+198*x^6+213*x^5-243*x^4-638*x^3+687*x^2+234*x+9) / (x-1)^11. - _Colin Barker_, Mar 22 2014

%t CoefficientList[Series[- 3 x^2 (40 x^8 - 185 x^7 + 198 x^6 + 213 x^5 - 243 x^4 - 638 x^3 + 687 x^2 + 234 x + 9)/(x - 1)^11, {x, 0, 40}], x] (* _Vincenzo Librandi_, Mar 23 2014 *)

%o (PARI) concat([0,0], Vec(-3*x^5*(40*x^8-185*x^7+198*x^6+213*x^5-243*x^4-638*x^3+687*x^2+234*x+9)/(x-1)^11 + O(x^100))) \\ _Colin Barker_, Mar 22 2014

%o (Magma) [(n^2-7*n+12)*(n^8+12*n^7-58*n^6-860*n^5+2141*n^4 +23728*n^3-61316*n^2-244928*n+770880)/3840: n in [3..40]]; // _Vincenzo Librandi_, Mar 23 2014

%Y Cf. A239567, A239575, A239568 (2 points), A239569 (3 points), A239570 (4 points), A282998 (6 points).

%K nonn,easy

%O 3,3

%A _Heinrich Ludwig_, Mar 22 2014