%I #14 Sep 05 2024 11:54:02

%S 0,1,90,1428,10600,51525,190806,584080,1552608,3701025,8088850,

%T 16470036,31616520,57743413,101055150,170433600,278290816,441610785,

%U 683206218,1033218100,1530887400,2226630021,3184447750,4484709648,6227340000,8535450625,11559457026,15481719540

%N Number of ways to place 4 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 5 of triangle A279445.

%C Rotations and reflections of placements are counted. For numbers if they are to be ignored see A279448.

%C For condition "no more than 2 points on straight lines at any angle", see A175383.

%H Heinrich Ludwig, <a href="/A279438/b279438.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (9,-36,84,-126,126,-84,36,-9,1).

%F a(n) = (n^8 - 14*n^6 + 30*n^5 - 17*n^4 - 6*n^3 + 6*n^2)/24.

%F a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9).

%F G.f.: x^2*(1 + 81*x + 654*x^2 + 904*x^3 + 99*x^4 - 57*x^5 - 2*x^6) / (1 - x)^9. - _Colin Barker_, Dec 13 2016

%t Table[(n^8 - 14 n^6 + 30 n^5 - 17 n^4 - 6 n^3 + 6 n^2)/24, {n, 28}] (* _Michael De Vlieger_, Dec 12 2016 *)

%t LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{0,1,90,1428,10600,51525,190806,584080,1552608},30] (* _Harvey P. Dale_, Sep 05 2024 *)

%o (PARI) concat(0, Vec(x^2*(1 + 81*x + 654*x^2 + 904*x^3 + 99*x^4 - 57*x^5 - 2*x^6) / (1 - x)^9 + O(x^30))) \\ _Colin Barker_, Dec 13 2016

%o (PARI) a(n) = (n^6 - 14*n^4 + 30*n^3 - 17*n^2 - 6*n + 6)*n^2/24 \\ _Charles R Greathouse IV_, Dec 13 2016

%Y Cf. A175383, A249448, A279444, A279445, A197458.

%Y Same problem but 2,3,5..9 points: A083374, A279437, A279439, A279440, A279441, A279442, A279443.

%K nonn,easy

%O 1,3

%A _Heinrich Ludwig_, Dec 12 2016