A279442 - OEIS

%I #14 Dec 24 2016 09:39:47

%S 0,0,0,90,67950,4531950,109425330,1460297160,13112872920,88456195800,

%T 480149029800,2196080372970,8743233946590,31033043111070,

%U 99992483914050,296626638016800,819218054279520,2125440234303840,5218743585428640,12201529135725450,27304286810701950

%N Number of ways to place 8 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 9 of triangle A279445.

%C Rotations and reflections of placements are counted.

%H Heinrich Ludwig, <a href="/A279442/b279442.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_17">Index entries for linear recurrences with constant coefficients</a>, signature (17,-136,680,-2380,6188,-12376,19448,-24310,24310,-19448,12376,-6188,2380,-680,136,-17,1).

%F a(n) = (n^16 -140*n^14 +756*n^13 +2506*n^12 -36540*n^11 +130940*n^10 -117432*n^9 -559615*n^8 +2186100*n^7 -3622360*n^6 +3228876*n^5 -1439892*n^4 +181440*n^3 +45360*n^2)/40320; factorized: a(n) = n^2*(n-1)^2*(n-2)^2*(n-3)^2*(n^8 +12*n^7 -54*n^6 -444*n^5 +1845*n^4 +1392*n^3 -11332*n^2 +9660*n +1260)/8!.

%F a(n) = SUM(1<=j<=17, C(17,j)*(-1)^(j-1)*a(n-j)).

%F G.f.: 90*x^4*(1 +738*x +37656*x^2 +461802*x^3 +1893555*x^4 +2767824*x^5 +1058064*x^6 -331920*x^7 -140913*x^8 +15950*x^9 +3096*x^10 -90*x^11 -3*x^12) / (1 -x)^17. - _Colin Barker_, Dec 23 2016

%t Table[n^2*(n - 1)^2*(n - 2)^2*(n - 3)^2*(n^8 + 12 n^7 - 54 n^6 - 444 n^5 + 1845 n^4 + 1392 n^3 - 11332 n^2 + 9660 n + 1260)/8!, {n, 21}] (* _Michael De Vlieger_, Dec 22 2016 *)

%o (PARI) concat(vector(3), Vec(90*x^4*(1 +738*x +37656*x^2 +461802*x^3 +1893555*x^4 +2767824*x^5 +1058064*x^6 -331920*x^7 -140913*x^8 +15950*x^9 +3096*x^10 -90*x^11 -3*x^12) / (1 -x)^17 + O(x^30))) \\ _Colin Barker_, Dec 23 2016

%Y Cf. A279444, A279445, A197458.

%Y Same problem but 2..7,9 points: A083374, A279437, A279438, A279439, A279440, A279441, A279443.

%K nonn,easy

%O 1,4

%A _Heinrich Ludwig_, Dec 22 2016