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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. 3
1, 3, 22, 188, 1712, 16098, 154352, 1499858, 14717692, 145509218, 1447187732, 14462966928, 145120265472, 1461040916988, 14751839744412, 149316973768398, 1514654852648052, 15393833895932658, 156716528008129892, 1597861126366223768 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Main diagonal of the square array given in A132439.

Recurrence relation: a_1=1; a_2=3; a_3=22; a_4=188; a_n=((29n-47)a_{n-1}+(-95n+238)a_{n-2}+(116n-418)a_{n-3}+(-48n+240)a_{n-4})/(2n-2), n >= 5. - Martin J. Erickson (erickson(AT)truman.edu), Nov 20 2007

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

REFERENCES

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. - Martin J. Erickson (erickson(AT)truman.edu), Oct 21 2010

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..300

FORMULA

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

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).

CROSSREFS

Cf. A132439.

Column k=2 of A229345.

Sequence in context: A077244 A138899 A147855 * 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

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Last modified September 17 01:33 EDT 2014. Contains 246831 sequences.