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%I #19 Dec 23 2016 21:43:47
%S 0,4,78,528,2200,6900,17934,40768,83808,159300,284350,482064,782808,
%T 1225588,1859550,2745600,3958144,5586948,7739118,10541200,14141400,
%U 18711924,24451438,31587648,40380000,51122500,64146654,79824528,98571928,120851700,147177150,178115584
%N Number of ways to place 3 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 4 of triangle A279445.
%C Rotations and reflections of placements are counted. For numbers if they are to be ignored see A279447.
%C For condition "no more than 2 points on straight lines at any angle", see A045996.
%H Heinrich Ludwig, <a href="/A279437/b279437.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F a(n) = (n^6 - 5*n^4 + 6*n^3 - 2*n^2)/6.
%F a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7).
%F G.f.: 2*x^2*(2 + 25*x + 33*x^2 + x^3 - x^4) / (1 - x)^7. - _Colin Barker_, Dec 12 2016
%t Table[(n^6 - 5 n^4 + 6 n^3 - 2 n^2)/6, {n, 32}] (* or *)
%t Rest@ CoefficientList[Series[2 x^2*(2 + 25 x + 33 x^2 + x^3 - x^4)/(1 - x)^7, {x, 0, 32}], x] (* _Michael De Vlieger_, Dec 12 2016 *)
%o (PARI) concat(0, Vec(2*x^2*(2 + 25*x + 33*x^2 + x^3 - x^4) / (1 - x)^7 + O(x^50))) \\ _Colin Barker_, Dec 12 2016
%Y Cf. A045996, A249447, A279444, A279445, A197458.
%Y Same problem but 2, 4..9 points: A083374, A279438, A279439, A279440, A279441, A279442, A279443.
%K nonn,easy
%O 1,2
%A _Heinrich Ludwig_, Dec 12 2016
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