A035005 Number of possible queen moves on an n X n chessboard.

0, 12, 56, 152, 320, 580, 952, 1456, 2112, 2940, 3960, 5192, 6656, 8372, 10360, 12640, 15232, 18156, 21432, 25080, 29120, 33572, 38456, 43792, 49600, 55900, 62712, 70056, 77952, 86420, 95480, 105152, 115456, 126412, 138040, 150360

Offset

1,2

Comments

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

Obviously A035005(n) = A002492(n-1) + A035006 (n) since Queen = Bishop + Rook. - Johannes W. Meijer, Feb 04 2010

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Milan Janjic and Boris Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - From N. J. A. Sloane, Feb 13 2013

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Formula

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

a(n) = 4 * A162147(n-1). - Johannes W. Meijer, Feb 04 2010

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

From Colin Barker, Mar 11 2012: (Start)

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

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

E.g.f.: 2*exp(x)*x^2*(9 + 5*x)/3. - Stefano Spezia, Jul 31 2022

Example

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

Mathematica

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 *)

Prog

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

Crossrefs

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

Cf. A162147.

Sequence in context: A340517 A104188 A069552 * A001386 A046998 A212507

Adjacent sequences: A035002 A035003 A035004 * A035006 A035007 A035008

Keyword

nonn,easy,nice

Author

Ulrich Schimke (ulrschimke(AT)aol.com)

Extensions

More terms from Erich Friedman

Status

approved