

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 upright.


3



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

1,2


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


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), 3748.


FORMULA

G.f.: (x*(x1)/(3*x2))*(1 + (1x)/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*n47)*a(n1) + (95*n + 238)*a(n2) + (116*n  418)*a(n3) + (48*n + 240)*a(n4))/(2*n2), n >= 5.  Martin J. Erickson (erickson(AT)truman.edu), Nov 20 2007


EXAMPLE

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).


MATHEMATICA

Rest[CoefficientList[Series[(x (x1)/(3x2))(1+(1x)/Sqrt[112x+16x^2]), {x, 0, 20}], x]] (* Harvey P. Dale, Feb 09 2015 *)


CROSSREFS

Cf. A132439.
Column k=2 of A229345.
Sequence in context: A138899 A147855 A278333 * A065204 A001393 A046743
Adjacent sequences: A132592 A132593 A132594 * A132596 A132597 A132598


KEYWORD

nonn,easy,nice


AUTHOR

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


STATUS

approved



