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A279449 Number of nonequivalent ways to place 5 points on an n X n square grid so that no more than 2 points are on a vertical or horizontal straight line.

%I A279449 #10 Dec 19 2016 18:36:59
%S A279449 0,0,9,306,4361,34059,185181,777280,2710074,8181558,22067973,54285858,
%T A279449 123791067,264749849,536146569,1035584592,1919530804,3430908108,
%U A279449 5937810417,9984193986,16358592141,26181281511,41019234245,63028246512,95136210222,141264963970,206611069197
%N A279449 Number of nonequivalent ways to place 5 points on an n X n square grid so that no more than 2 points are on a vertical or horizontal straight line.
%C A279449 Column 6 of A279453.
%C A279449 Rotations and reflections of placements are not counted. For numbers if they are to be counted see A279439.
%C A279449 For condition "no more than 2 points on straight lines at any angle", see A235456.
%H A279449 Heinrich Ludwig, <a href="/A279449/b279449.txt">Table of n, a(n) for n = 1..1000</a>
%H A279449 <a href="/index/Rec#order_17">Index entries for linear recurrences with constant coefficients</a>, signature (5,-4,-20,40,16,-100,44,110,-110,-44,100,-16,-40,20,4,-5,1).
%F A279449 a(n) = (n^10 - 30*n^8 + 90*n^7 - 27*n^6 - 218*n^5 + 340*n^4 - 340*n^3 + 376*n^2 - 192*n)/960 + IF(MOD(n, 2) = 1, 2*n^5 - 9*n^4 + 14*n^3 - 6*n^2 - 4*n + 3)/64.
%F A279449 a(n) = 5*a(n-1) - 4*a(n-2) - 20*a(n-3) + 40*a(n-4) + 16*a(n-5) - 100*a(n-6) + 44*a(n-7) + 110*a(n-8) - 110*a(n-9) - 44*a(n-10) + 100*a(n-11) - 16*a(n-12) - 40*a(n-13) + 20*a(n-14) + 4*a(n-15) - 5*a(n-16) + a(n-17).
%F A279449 G.f.: x^3*(9 +261*x +2867*x^2 +13658*x^3 +38090*x^4 +62447*x^5 +67142*x^6 +41996*x^7 +15541*x^8 +955*x^9 -761*x^10 -278*x^11 -8*x^12 +x^13) / ((1 -x)^11*(1 +x)^6). - _Colin Barker_, Dec 18 2016
%o A279449 (PARI) concat(vector(2), Vec(x^3*(9 +261*x +2867*x^2 +13658*x^3 +38090*x^4 +62447*x^5 +67142*x^6 +41996*x^7 +15541*x^8 +955*x^9 -761*x^10 -278*x^11 -8*x^12 +x^13) / ((1 -x)^11*(1 +x)^6) + O(x^40))) \\ _Colin Barker_, Dec 18 2016
%Y A279449 Cf. A235456, A279439, A279452, A279453, A279454.
%Y A279449 Same problem but 2,3,4,6,7 points: A014409, A279447, A279448, A279450, A279451.
%K A279449 nonn,easy
%O A279449 1,3
%A A279449 _Heinrich Ludwig_, Dec 18 2016