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A296996 Number of nonequivalent (mod D_8) ways to place 3 points on an n X n point grid so that no point is equally distant from two other points on the same row or the same column.

%I A296996 #9 Jan 16 2018 03:15:31
%S A296996 0,1,14,75,310,911,2373,5254,10824,20305,36300,61081,99294,154735,
%T A296996 234955,345836,498848,702609,973674,1324135,1776950,2348511,3069649,
%U A296996 3961970,5065800,6408961,8043048,10003189,12354174,15139615,18439575,22307416,26840704,32103905,38214470
%N A296996 Number of nonequivalent (mod D_8) ways to place 3 points on an n X n point grid so that no point is equally distant from two other points on the same row or the same column.
%C A296996 Rotations and reflections of placements are not counted. If they are to be counted see A296997.
%C A296996 The condition of placements is also known as "no 3-term arithmetic progressions".
%H A296996 Heinrich Ludwig, <a href="/A296996/b296996.txt">Table of n, a(n) for n = 1..1000</a>
%H A296996 <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (3,1,-11,6,14,-14,-6,11,-1,-3,1).
%F A296996 a(n) = (n^6 -3*n^4 +5*n^3 -4*n^2 +4n)/48 + (n == 1 mod 2)*(8*n^3 -18n^2 +7*n)/48.
%F A296996 From _Colin Barker_, Jan 12 2018: (Start)
%F A296996 G.f.: x^2*(1 + 11*x + 32*x^2 + 82*x^3 + 54*x^4 + 57*x^5 + 2*x^6 + 2*x^7 - x^8) / ((1 - x)^7*(1 + x)^4).
%F A296996 a(n) = (n^6 - 3*n^4 + 5*n^3 - 4*n^2 + 4*n) / 48 for n even.
%F A296996 a(n) = (n^6 - 3*n^4 + 13*n^3 - 22*n^2 + 11*n) / 48 for n odd.
%F A296996 a(n) = 3*a(n-1) + a(n-2) - 11*a(n-3) + 6*a(n-4) + 14*a(n-5) - 14*a(n-6) - 6*a(n-7) + 11*a(n-8) - a(n-9) - 3*a(n-10) + a(n-11) for n>11.
%F A296996 (End)
%t A296996 Array[(#^6 - 3 #^4 + 5 #^3 - 4 #^2 + 4 #)/48 + Boole[OddQ@ #] (8 #^3 - 18 #^2 + 7 #)/48 &, 35] (* or *)
%t A296996 Rest@ CoefficientList[Series[x^2*(1 + 11 x + 32 x^2 + 82 x^3 + 54 x^4 + 57 x^5 + 2 x^6 + 2 x^7 - x^8)/((1 - x)^7*(1 + x)^4), {x, 0, 35}], x] (* _Michael De Vlieger_, Jan 12 2018 *)
%o A296996 (PARI) concat(0, Vec(x^2*(1 + 11*x + 32*x^2 + 82*x^3 + 54*x^4 + 57*x^5 + 2*x^6 + 2*x^7 - x^8) / ((1 - x)^7*(1 + x)^4) + O(x^40))) \\ _Colin Barker_, Jan 12 2018
%Y A296996 Cf. A296997.
%K A296996 nonn,easy
%O A296996 1,3
%A A296996 _Heinrich Ludwig_, Jan 12 2018