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Array[(#^6 - 3 #^4 + 5 #^3 - 4 #^2 + 4 #)/48 + Boole[OddQ@ #] (8 #^3 - 18 #^2 + 7 #)/48 &, 35] (* or *)
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 *)
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<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).
From Colin Barker, Jan 12 2018: (Start)
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).
a(n) = (n^6 - 3*n^4 + 5*n^3 - 4*n^2 + 4*n) / 48 for n even.
a(n) = (n^6 - 3*n^4 + 13*n^3 - 22*n^2 + 11*n) / 48 for n odd.
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.
(End)
(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
nonn,changed,easy
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allocated for Heinrich LudwigNumber 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.
0, 1, 14, 75, 310, 911, 2373, 5254, 10824, 20305, 36300, 61081, 99294, 154735, 234955, 345836, 498848, 702609, 973674, 1324135, 1776950, 2348511, 3069649, 3961970, 5065800, 6408961, 8043048, 10003189, 12354174, 15139615, 18439575, 22307416, 26840704, 32103905, 38214470
1,3
Rotations and reflections of placements are not counted. If they are to be counted see A296997.
The condition of placements is also known as "no 3-term arithmetic progressions".
Heinrich Ludwig, <a href="/A296996/b296996.txt">Table of n, a(n) for n = 1..1000</a>
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.
Cf. A296997.
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Heinrich Ludwig, Jan 12 2018
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