%I #12 Dec 19 2016 18:38:25

%S 0,0,2,285,9110,126396,1055025,6266614,29198740,113262680,380775248,

%T 1140764611,3108667306,7824370092,18407341855,40855872764,86201399496,

%U 173952773328,337453762782,631982899545,1146743732126,2022212701212,3474824082125,5831439251154,9576836632860

%N Number of nonequivalent ways to place 6 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 7 of A279453.

%C Rotations and reflections of placements are not counted. For numbers if they are to be counted see A279440.

%C For condition "no more than 2 points on straight lines at any angle", see A235457.

%H Heinrich Ludwig, <a href="/A279450/b279450.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_20">Index entries for linear recurrences with constant coefficients</a>, signature (6,-8,-22,69,-8,-176,168,182,-364,0,364,-182,-168,176,8,-69,22,8,-6,1).

%F a(n) = (n^12 - 55*n^10 + 210*n^9 + 93*n^8 - 2220*n^7 + 6052*n^6 - 8040*n^5 + 4236*n^4 + 3240*n^3 - 5872*n^2 + 2400*n)/5760 + IF(MOD(n, 2) = 1, 2*n^6 - 18*n^5 + 53*n^4 - 64*n^3 + 33*n^2 - 12*n + 5)/128.

%F a(n) = 6*a(n-1) - 8*a(n-2) - 22*a(n-3) + 69*a(n-4) - 8*a(n-5) - 176*a(n-6) + 168*a(n-7) + 182*a(n-8) - 364*a(n-9) + 364*a(n-11) - 182*a(n-12) - 168*a(n-13) + 176*a(n-14) + 8*a(n-15) - 69*a(n-16) + 22*a(n-17) + 8*a(n-18) - 6*a(n-19) + *a(n-20).

%F G.f.: x^3*(2 +273*x +7416*x^2 +74060*x^3 +375661*x^4 +1128403*x^5 +2194010*x^6 +2815082*x^7 +2424155*x^8 +1294751*x^9 +376028*x^10 -5296*x^11 -32173*x^12 -8195*x^13 +178*x^14 +122*x^15 +3*x^16) / ((1 -x)^13*(1 +x)^7). - _Colin Barker_, Dec 18 2016

%o (PARI) concat(vector(2), Vec(x^3*(2 +273*x +7416*x^2 +74060*x^3 +375661*x^4 +1128403*x^5 +2194010*x^6 +2815082*x^7 +2424155*x^8 +1294751*x^9 +376028*x^10 -5296*x^11 -32173*x^12 -8195*x^13 +178*x^14 +122*x^15 +3*x^16) / ((1 -x)^13*(1 +x)^7) + O(x^40))) \\ _Colin Barker_, Dec 18 2016

%Y Cf. A235457, A279440, A279452, A279453, A279454.

%Y Same problem but 2,3,4,5,7 points: A014409, A279447, A279448, A279449, A279451.

%K nonn,easy

%O 1,3

%A _Heinrich Ludwig_, Dec 18 2016