A035006 Number of possible rook moves on an n X n chessboard.

0, 8, 36, 96, 200, 360, 588, 896, 1296, 1800, 2420, 3168, 4056, 5096, 6300, 7680, 9248, 11016, 12996, 15200, 17640, 20328, 23276, 26496, 30000, 33800, 37908, 42336, 47096, 52200, 57660, 63488, 69696, 76296, 83300, 90720, 98568, 106856

Offset

1,2

Comments

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

X values of solutions of the equation: (X-Y)^3-2*X*Y=0. Y values are b(n)=2*n*(n-1)^2 (see A181617). - Mohamed Bouhamida, Jul 06 2023

References

E. Bonsdorff, K. Fabel and O. Riihimaa, Schach und Zahl (Chess and numbers), Walter Rau Verlag, Dusseldorf, 1966.

Links

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

Alexander M. Haupt, Bijective enumeration of rook walks, arXiv:2007.01018 [math.CO], 2020.

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

M. Janjic and B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5.

Richard P. Stanley, Bijective Proof Problems, Problem 540 p. 63, (2015).

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

Formula

a(n) = (n-1)*2*n^2.

a(n) = Sum_{j=1..n} ((n+j-1)^2 - (n-j+1)^2). - Zerinvary Lajos, Sep 13 2006

1/a(n+1) = Integral_{x=1/(n+1)..1/n} x*h(x) = Integral_{x=1/(n+1)..1/n} x*(1/x - floor(1/x)) = 1/((2*(n^2+2*n+1))*n) and Sum_{n>=1} 1/((2*(n^2+2*n+1))*n) = 1-Zeta(2)/2 where h(x) is the Gauss (continued fraction) map h(x)={x^-1} and {x} is the fractional part of x. - Stephen Crowley, Jul 24 2009

a(n) = 4 * A006002(n-1). - Johannes W. Meijer, Feb 04 2010

G.f.: 4*x^2*(2+x)/(1-x)^4. - Colin Barker, Mar 11 2012

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(1)=0, a(2)=8, a(3)=36, a(4)=96. - Harvey P. Dale, May 12 2012

a(n) = A006566(n) - A006564(n). - Peter M. Chema, Feb 10 2016

E.g.f.: 2*exp(x)*x^2*(2 + x). - Stefano Spezia, May 10 2022

From Amiram Eldar, May 14 2022: (Start)

Sum_{n>=2} 1/a(n) = 1 - Pi^2/12.

Sum_{n>=2} (-1)^n/a(n) = Pi^2/24 + log(2) - 1. (End)

Example

On a 3 X 3-board, rook has 9*4 moves, so a(3)=36.

Mathematica

Table[(n-1) 2 n^2, {n, 40}] (* or *) LinearRecurrence[{4, -6, 4, -1}, {0, 8, 36, 96}, 40] (* Harvey P. Dale, May 12 2012 *)

Prog

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

Crossrefs

Cf. A033586 (King), A035005 (Queen), A035008 (Knight), A002492 (Bishop) and A049450 (Pawn).

Cf. A006002, A006564, A006566.

Sequence in context: A187287 A224159 A321251 * A245360 A032768 A224410

Adjacent sequences: A035003 A035004 A035005 * A035007 A035008 A035009

Keyword

easy,nonn,nice

Author

Ulrich Schimke (ulrschimke(AT)aol.com)

Status

approved