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A132595
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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.
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3
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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
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1,2
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COMMENTS
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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
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REFERENCES
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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
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 1..300
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FORMULA
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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).
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EXAMPLE
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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).
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MATHEMATICA
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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 *)
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CROSSREFS
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Cf. A132439.
Column k=2 of A229345.
Sequence in context: A077244 A138899 A147855 * A065204 A001393 A046743
Adjacent sequences: A132592 A132593 A132594 * A132596 A132597 A132598
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KEYWORD
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nonn,easy,nice
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AUTHOR
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Martin J. Erickson (erickson(AT)truman.edu), Nov 14 2007, Jan 28 2009
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STATUS
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approved
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