A243717 - OEIS

%I #19 Sep 08 2022 08:46:08

%S 2,7,18,43,83,156,257,418,624,925,1292,1797,2393,3178,4083,5236,6542,

%T 8163,9974,12175,14607,17512,20693,24438,28508,33241,38352,44233,

%U 50549,57750,65447,74152,83418,93823,104858,117171,130187,144628,159849,176650,194312

%N Number of inequivalent (mod D_4) ways to place 2 nonattacking knights on an n X n board.

%C Rotations or reflections of a placement are considered as the same. If they are distinguished, numbers are A172132.

%H Heinrich Ludwig, <a href="/A243717/b243717.txt">Table of n, a(n) for n = 2..1000</a>

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-6,0,6,-2,-2,1).

%F a(n) = (n^4 - 2*n^2 + 20*n - 16 + IF(MOD(n, 2) = 1)*(2*n^2 - 4*n - 1))/16.

%F a(n) = (-33+(-1)^n+4*(9+(-1)^n)*n-2*(1+(-1)^n)*n^2+2*n^4)/32. - _Colin Barker_, Jun 10 2014

%F G.f.: x^2*(x^6-3*x^4-5*x^3-3*x-2) / ((x-1)^5*(x+1)^3). - _Colin Barker_, Jun 10 2014

%p A243717:=n->(-33+(-1)^n+4*(9+(-1)^n)*n-2*(1+(-1)^n)*n^2+2*n^4)/32; seq(A243717(n), n=2..50); # _Wesley Ivan Hurt_, Jun 11 2014

%t Table[(-33 + (-1)^n + 4*(9 + (-1)^n)*n - 2*(1 + (-1)^n)*n^2 + 2*n^4)/

%t 32, {n, 2, 50}] (* _Wesley Ivan Hurt_, Jun 11 2014 *)

%o (PARI) Vec(x^2*(x^6-3*x^4-5*x^3-3*x-2)/((x-1)^5*(x+1)^3) + O(x^100)) \\ _Colin Barker_, Jun 10 2014

%o (Magma) [ (-33+(-1)^n+4*(9+(-1)^n)*n-2*(1+(-1)^n)*n^2+2*n^4)/32: n in [2..50]]; // _Wesley Ivan Hurt_, Jun 11 2014

%Y Cf. A243716, A172132, A243718, A243719, A243720.

%K nonn,easy

%O 2,1

%A _Heinrich Ludwig_, Jun 10 2014