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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).

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 *)

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

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Last modified December 9 06:57 EST 2016. Contains 278963 sequences.