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Number of possible queen moves on an n X n chessboard.
9

%I #49 Aug 01 2022 08:11:41

%S 0,12,56,152,320,580,952,1456,2112,2940,3960,5192,6656,8372,10360,

%T 12640,15232,18156,21432,25080,29120,33572,38456,43792,49600,55900,

%U 62712,70056,77952,86420,95480,105152,115456,126412,138040,150360

%N Number of possible queen moves on an n X n chessboard.

%C The number of (2 to n) digit sequences that can be found reading in any orientation, including diagonals, in an (n X n) grid. - _Paul Cleary_, Aug 12 2005

%C Obviously A035005(n) = A002492(n-1) + A035006 (n) since Queen = Bishop + Rook. - _Johannes W. Meijer_, Feb 04 2010

%H Vincenzo Librandi, <a href="/A035005/b035005.txt">Table of n, a(n) for n = 1..1000</a>

%H Milan Janjic and Boris Petkovic, <a href="http://arxiv.org/abs/1301.4550">A Counting Function</a>, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - From _N. J. A. Sloane_, Feb 13 2013

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

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

%F a(n) = 4 * A162147(n-1). - _Johannes W. Meijer_, Feb 04 2010

%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(0)=0, a(1)=12, a(2)=56, a(3)=152. - _Harvey P. Dale_, Aug 24 2011

%F From _Colin Barker_, Mar 11 2012: (Start)

%F a(n) = 2*n*(1-6*n+5*n^2)/3.

%F G.f.: 4*x^2*(3+2*x)/(1-x)^4. (End)

%F E.g.f.: 2*exp(x)*x^2*(9 + 5*x)/3. - _Stefano Spezia_, Jul 31 2022

%e 3 X 3 board: queen has 8*6 moves and 1*8 moves, so a(3)=56.

%t Table[(n-1)2n^2+(4n^3-6n^2+2n)/3,{n,40}] (* or *) LinearRecurrence[ {4,-6,4,-1},{0,12,56,152},40] (* _Harvey P. Dale_, Aug 24 2011 *)

%o (Magma) [(n-1)*2*n^2 + (4*n^3-6*n^2+2*n)/3: n in [1..40]]; // _Vincenzo Librandi_, Jun 16 2011

%Y Cf. A033586 (King), A035006 (Rook), A035008 (Knight), A002492 (Bishop) and A049450 (Pawn).

%Y Cf. A162147.

%K nonn,easy,nice

%O 1,2

%A Ulrich Schimke (ulrschimke(AT)aol.com)

%E More terms from _Erich Friedman_