%I #17 Sep 08 2022 08:46:18
%S 0,1,14,73,301,890,2321,5166,10654,20055,35880,60511,98419,153608,
%T 233331,343820,496076,699261,969234,1318885,1770185,2340646,3059749,
%U 3950618,5051786,6393075,8023756,9981531,12328239,15110740,18405415,22269656,26796504,32055353,38158166
%N Number of nonequivalent ways to place 3 points on an n X n square grid so that no more than 2 points are on a vertical or horizontal straight line.
%C Column 4 of A279453.
%C Rotations and reflections of placements are not counted. For numbers if they are to be counted see A279437.
%C For condition "no more than 2 points on straight lines at any angle", see A235454.
%H Heinrich Ludwig, <a href="/A279447/b279447.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (3,1,-11,6,14,-14,-6,11,-1,-3,1).
%F a(n) = (n^6 - 5*n^4 + 14*n^3 - 14*n^2 + 4*n)/48 + IF(MOD(n, 2) = 1, 2*n^3 - 3*n^2 + 1)/16.
%F a(n) = 3*a(n-1) + a(n-2) - 11*a(n-3) + 6*a(n-4) + 14*a(n-5) - 14*a(n-6) - 6*a(n-7) + 11*a(n-8) - a(n-9) - 3*a(n-10) + a(n-11).
%F G.f.: x^2*(1 + 11*x + 30*x^2 + 79*x^3 + 62*x^4 + 55*x^5 + 4*x^6 - x^7 - x^8) / ((1 - x)^7*(1 + x)^4). - _Colin Barker_, Dec 17 2016
%t LinearRecurrence[{3, 1, -11, 6, 14, -14, -6, 11, -1, -3, 1},{0, 1, 14, 73, 301, 890, 2321, 5166, 10654, 20055, 35880}, 35] (* _Vincenzo Librandi_ Dec 17 2016 *)
%o (Magma) I:=[0,1,14,73,301,890,2321,5166,10654,20055,35880]; [n le 11 select I[n] else 3*Self(n-1)+Self(n-2)-11*Self(n-3)+ 6*Self(n-4)+14*Self(n-5)-14*Self(n-6)-6*Self(n-7)+11*Self(n-8)-Self(n-9)-3*Self(n-10)+Self(n-11): n in [1..40]]; // _Vincenzo Librandi_, Dec 17 2016
%o (PARI) concat(0, Vec(x^2*(1 + 11*x + 30*x^2 + 79*x^3 + 62*x^4 + 55*x^5 + 4*x^6 - x^7 - x^8) / ((1 - x)^7*(1 + x)^4) + O(x^30))) \\ _Colin Barker_, Dec 17 2016
%Y Cf. A235454, A279437, A279452, A279453, A279454.
%Y Same problem but 2, 4..7 points: A014409, A279448, A279449, A279450, A279451.
%K nonn,easy
%O 1,3
%A _Heinrich Ludwig_, Dec 17 2016