A132595 Number of ways to move a chess queen from the lower left corner to square (n,n), with the queen moving only up, right, or diagonally up-right.

1, 3, 22, 188, 1712, 16098, 154352, 1499858, 14717692, 145509218, 1447187732, 14462966928, 145120265472, 1461040916988, 14751839744412, 149316973768398, 1514654852648052, 15393833895932658, 156716528008129892, 1597861126366223768

Offset: 1

Author

Martin J. Erickson (erickson(AT)truman.edu), Nov 14 2007, Jan 28 2009

Keywords

• nonn
• easy
• nice

Comments

Main diagonal of the square array given in A132439.

a(n) is the number of Wythoff's Nim games starting with two equal piles of n stones. - Martin J. Erickson (erickson(AT)truman.edu), Dec 05 2008

Examples

			a(2) = 3 since the paths from (1,1) to (2,2) are
(1,1)->(2,1)->(2,2),
(1,1)->(1,2)->(2,2),
(1,1)->(2,2).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..300
• M. Erickson, S. Fernando, K. Tran, Enumerating Rook and Queen Paths , Bulletin of the Institute for Combinatorics and Its Applications, Volume 60 (2010), 37-48.

Crossrefs

Cf. A132439.

Column k=2 of A229345.

Programs

Mathematica

Rest[CoefficientList[Series[(x (x-1)/(3x-2))(1+(1-x)/Sqrt[1-12x+16x^2]),{x,0,20}],x]] (* Harvey P. Dale, Feb 09 2015 *)

Formula

G.f.: (x*(x-1)/(3*x-2))*(1 + (1-x)/sqrt(1 - 12*x + 16*x^2)). a(n) is asymptotic to (5^(3/4)*(sqrt(5)-2)/16)*(6+2*sqrt(5))^n/sqrt(Pi*n).

a(1)=1; a(2)=3; a(3)=22; a(4)=188; a(n) = ((29*n-47)*a(n-1) + (-95*n + 238)*a(n-2) + (116*n - 418)*a(n-3) + (-48*n + 240)*a(n-4))/(2*n-2), n >= 5. - Martin J. Erickson (erickson(AT)truman.edu), Nov 20 2007