%I #28 Aug 08 2023 17:12:57

%S 0,1,21,151,615,1845,4571,9926,19566,35805,61765,101541,160381,244881,

%T 363195,525260,743036,1030761,1405221,1886035,2495955,3261181,4211691,

%U 5381586,6809450,8538725,10618101,13101921,16050601,19531065,23617195,28390296,33939576

%N Number of ways to place 3 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 A239573.

%H Vincenzo Librandi, <a href="/A239569/b239569.txt">Table of n, a(n) for n = 2..1000</a>

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1)

%F a(n) = (n-1)*(n-2)*(n^4+6*n^3-23*n^2-92*n+264)/48.

%F G.f.: -x^3*(11*x^4-36*x^3+25*x^2+14*x+1) / (x-1)^7. - _Colin Barker_, Mar 22 2014

%t CoefficientList[Series[- x (11 x^4 - 36 x^3 + 25 x^2 + 14 x + 1)/(x - 1)^7, {x, 0, 40}], x] (* _Vincenzo Librandi_, Mar 23 2014 *)

%t LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,1,21,151,615,1845,4571},50] (* _Harvey P. Dale_, Aug 08 2023 *)

%o (PARI) concat(0, Vec(-x^3*(11*x^4-36*x^3+25*x^2+14*x+1)/(x-1)^7 + O(x^100))) \\ _Colin Barker_, Mar 22 2014

%o (Magma) [(n^2-3*n+2)*(n^4+6*n^3-23*n^2-92*n+264)/48: n in [2..40]]; // _Vincenzo Librandi_, Mar 23 2014

%Y Cf. A239567, A239573, A239568 (2 points), A239570 (4 points), A239571 (5 points), A282998 (6 points).

%K nonn,easy

%O 2,3

%A _Heinrich Ludwig_, Mar 22 2014