A279448 - OEIS

%I #11 Dec 19 2016 05:45:50

%S 0,1,17,202,1397,6582,24185,73496,195086,463875,1013505,2061426,

%T 3956947,7222992,12640817,21312992,34801420,55215621,85424721,

%U 129174250,191397185,278361226,398108777,560635032,778491962,1066995527,1445034305,1935301746,2565356031,3367870500

%N Number of nonequivalent 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 A279453.

%C Rotations and reflections of placements are not counted. For numbers if they are to be counted see A279438.

%C For condition "no more than 2 points on straight lines at any angle", see A235455.

%H Heinrich Ludwig, <a href="/A279448/b279448.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_14">Index entries for linear recurrences with constant coefficients</a>, signature (4,-1,-16,19,20,-45,0,45,-20,-19,16,1,-4,1).

%F a(n) = (n^8 - 14*n^6 + 30*n^5 + 12*n^4 - 60*n^3 + 40*n^2)/192 + IF(MOD(n, 2) = 1, 4*n^4 - 20*n^3 + 22*n^2 - 2*n - 7)/64.

%F a(n) = 4*a(n-1) - a(n-2) - 16*a(n-3) + 19*a(n-4) + 20*a(n-5) - 45*a(n-6) + 45*a(n-8) - 20*a(n-9) - 19*a(n-10) + 16*a(n-11) + a(n-12) - 4*a(n-13) + a(n-14).

%F G.f.: x^2*(1 +13*x +135*x^2 +622*x^3 +1449*x^4 +2143*x^5 +1557*x^6 +781*x^7 +34*x^8 -8*x^9 -8*x^10 +x^11) / ((1 -x)^9*(1 +x)^5). - _Colin Barker_, Dec 18 2016

%o (PARI) concat(0, Vec(x^2*(1 +13*x +135*x^2 +622*x^3 +1449*x^4 +2143*x^5 +1557*x^6 +781*x^7 +34*x^8 -8*x^9 -8*x^10 +x^11) / ((1 -x)^9*(1 +x)^5) + O(x^40))) \\ _Colin Barker_, Dec 18 2016

%Y Cf. A235455, A279438, A279452, A279453, A279454.

%Y Same problem but 2,3,5,6,7 points: A014409, A279447, A279449, A279450, A279451.

%K nonn,easy

%O 1,3

%A _Heinrich Ludwig_, Dec 18 2016