cp's OEIS Frontend

A279451 Number of nonequivalent ways to place 7 points on an n X n square grid so that no more than 2 points are on a vertical or horizontal straight line.

%I A279451 #9 Dec 23 2016 21:44:26
%S A279451 0,0,0,115,11810,326190,4444935,38675954,246563232,1248782460,
%T A279451 5296300670,19499431941,63958228738,190528987506,523151460045,
%U A279451 1339408935540,3227223506896,7372750196952,16069268866908,33586411339335,67610793877650,131569779776182,248290280743571
%N A279451 Number of nonequivalent ways to place 7 points on an n X n square grid so that no more than 2 points are on a vertical or horizontal straight line.
%C A279451 Column 8 of A279453.
%C A279451 Rotations and reflections of placements are not counted. For numbers if they are to be counted see A279441.
%C A279451 For condition "no more than 2 points on straight lines at any angle", see A235458.
%H A279451 Heinrich Ludwig, <a href="/A279451/b279451.txt">Table of n, a(n) for n = 1..1000</a>
%F A279451 a(n) = (n^14 -91*n^12 +420*n^11 +693*n^10 -10500*n^9 +33647*n^8 -45316*n^7 +3682*n^6 +62300*n^5 -51996*n^4 -28504*n^3 +54384*n^2 -18720*n)/40320 + IF(MOD(n, 2) = 1, 2*n^7 -17*n^6 +50*n^5 -59*n^4 +38*n^3 -71*n^2 +102*n -45)/384.
%F A279451 G.f.: x^4*(115 +11005*x +245015*x^2 +2317550*x^3 +12037814*x^4 +39232894*x^5 +85494738*x^6 +129182670*x^7 +135873108*x^8 +97856368*x^9 +44499480*x^10 +9709722*x^11 -1359254*x^12 -1352974*x^13 -257282*x^14 +13866*x^15 +7705*x^16 +419*x^17 +x^18) / ((1 -x)^15*(1 +x)^8). - _Colin Barker_, Dec 23 2016
%t A279451 Table[(n^14 - 91 n^12 + 420 n^11 + 693 n^10 - 10500 n^9 + 33647 n^8 - 45316 n^7 + 3682 n^6 + 62300 n^5 - 51996 n^4 - 28504 n^3 + 54384 n^2 - 18720 n)/40320 + Boole[OddQ@ n] (2 n^7 - 17 n^6 + 50 n^5 - 59 n^4 + 38 n^3 - 71 n^2 + 102 n - 45)/384, {n, 23}] (* or *)
%t A279451 Rest@ CoefficientList[Series[x^4*(115 + 11005 x + 245015 x^2 + 2317550 x^3 + 12037814 x^4 + 39232894 x^5 + 85494738 x^6 + 129182670 x^7 + 135873108 x^8 + 97856368 x^9 + 44499480 x^10 + 9709722 x^11 - 1359254 x^12 - 1352974 x^13 - 257282 x^14 + 13866 x^15 + 7705 x^16 + 419 x^17 + x^18)/((1 - x)^15*(1 + x)^8), {x, 0, 23}], x] (* _Michael De Vlieger_, Dec 23 2016 *)
%o A279451 (PARI) concat(vector(3), Vec(x^4*(115 +11005*x +245015*x^2 +2317550*x^3 +12037814*x^4 +39232894*x^5 +85494738*x^6 +129182670*x^7 +135873108*x^8 +97856368*x^9 +44499480*x^10 +9709722*x^11 -1359254*x^12 -1352974*x^13 -257282*x^14 +13866*x^15 +7705*x^16 +419*x^17 +x^18) / ((1 -x)^15*(1 +x)^8) + O(x^30))) \\ _Colin Barker_, Dec 23 2016
%Y A279451 Cf. A235458, A279441, A279452, A279453, A279454.
%Y A279451 Same problem but 2..6 points: A014409, A279447, A279448, A279449, A279450.
%K A279451 nonn,easy
%O A279451 1,4
%A A279451 _Heinrich Ludwig_, Dec 22 2016