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A296997 Number of 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. 3

%I

%S 0,4,78,544,2260,7068,18298,41472,85032,161300,287430,486624,789308,

%T 1234604,1871730,2761728,3979088,5613732,7772862,10583200,14193060,

%U 18774844,24527338,31678464,40487800,51249588,64295478,79997792,98772492,121082700,147441890,178417664

%N Number of 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 Rotations and reflections of a placement are counted. If they are to be ignored, see A296996.

%C The condition of placements is also known as "no 3-term arithmetic progressions".

%H Heinrich Ludwig, <a href="/A296997/b296997.txt">Table of n, a(n) for n = 1..256</a>

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (5,-8,0,14,-14,0,8,-5,1).

%F a(n) = (n^6 - 3*n^4 - 3*n^3 + 8*n^2)/6 - (n == 1 (mod 2))*n/2.

%F a(n) = (n^6 - 3*n^4 - 3*n^3 + 8*n^2)/6 for n even,

%F a(n) = (n^6 - 3*n^4 - 3*n^3 + 8*n^2 - 3*n)/6 for n odd.

%F From _Colin Barker_, Dec 23 2017: (Start)

%F G.f.: 2*x^2*(2 + 29*x + 93*x^2 + 82*x^3 + 32*x^4 + x^5 + x^6) / ((1 - x)^7*(1 + x)^2).

%F a(n) = 5*a(n-1) - 8*a(n-2) + 14*a(n-4) - 14*a(n-5) + 8*a(n-7) - 5*a(n-8) + a(n-9) for n>9.

%F (End)

%t Array[(#^6 - 3 #^4 - 3 #^3 + 8 #^2)/6 - # Boole[OddQ@ #]/2 &, 32] (* _Michael De Vlieger_, Dec 23 2017 *)

%t CoefficientList[ Series[-2x (2 + 29x + 93x^2 + 82x^3 + 32x^4 + x^5 + x^6)/((x - 1)^7 (x + 1)^2), {x, 0, 31}], x] (* or *)

%t LinearRecurrence[{5, -8, 0, 14, -14, 0, 8, -5, 1}, {0, 4, 78, 544, 2260, 7068, 18298, 41472, 85032}, 32] (* _Robert G. Wilson v_, Jan 15 2018 *)

%o (PARI) concat(0, Vec(2*x^2*(2 + 29*x + 93*x^2 + 82*x^3 + 32*x^4 + x^5 + x^6) / ((1 - x)^7*(1 + x)^2) + O(x^40))) \\ _Colin Barker_, Dec 23 2017

%Y Cf. A296468, A296996, A296998.

%K nonn,easy

%O 1,2

%A _Heinrich Ludwig_, Dec 23 2017

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