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A057648
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Number of excursions of length n on the upper-right part of the hexagonal lattice.
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1
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1, 0, 2, 2, 13, 34, 158, 594, 2665, 11558, 53320, 247488, 1181266, 5708884, 28049474, 139417402, 701063005, 3559326294, 18233244530, 94140532624, 489573775236, 2562613997512, 13493827469116, 71441865994904
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OFFSET
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0,3
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COMMENTS
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Excursions = walks from the origin to the origin.
The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice. - Sean A. Irvine, Jun 22 2022
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LINKS
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FORMULA
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G.f.: (1-2*x)*hypergeom([-1/2, 1/2],[2],16*x^2/(1-2*x)^2)/(4*x^2) - (2*x+1)*((1-6*x)*hypergeom([1/3, 2/3],[2],27*x^2*(2*x+1))+1/2)/(6*x^2). - Mark van Hoeij, Dec 08 2014
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MAPLE
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gf:=(1-2*x)*hypergeom([-1/2, 1/2], [2], 16*x^2/(1-2*x)^2)/(4*x^2) - (2*x+1)*((1-6*x)*hypergeom([1/3, 2/3], [2], 27*x^2*(2*x+1))+1/2)/(6*x^2):
S:= series(gf, x, 103):
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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