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A286760
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Total number of nodes summed over all lattice paths from (0,0) to (n,0) that do not go below the x-axis or above the diagonal x=y and consist of steps U=(1,1), D=(1,-1), H=(1,0) and S=(0,1).
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2
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1, 2, 10, 42, 214, 1098, 5978, 33190, 189078, 1093490, 6414714, 38027030, 227489950, 1370980490, 8314674202, 50696630838, 310541818382, 1909850054666, 11786947172234, 72969941803662, 452976340653030, 2818815920369754, 17579546535174946, 109850944544149134
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OFFSET
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0,2
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LINKS
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FORMULA
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a(n) ~ c * (3 + 2*sqrt(3))^n / sqrt(n), where c = 0.0889843039487036085233000284915570190371055498671732340656... - Vaclav Kotesovec, Sep 11 2021
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MAPLE
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b:= proc(x, y) option remember; `if`(y<0 or y>x, 0, `if`(x=0, [1$2],
(p-> p+[0, p[1]])(b(x-1, y)+b(x, y-1)+b(x-1, y-1)+b(x-1, y+1))))
end:
a:= n-> b(n, 0)[2]:
seq(a(n), n=0..30);
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MATHEMATICA
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b[x_, y_] := b[x, y] = If[y<0 || y>x, 0, If[x == 0, {1, 1}, Function[
p, p+{0, p[[1]]}][b[x-1, y] + b[x, y-1] + b[x-1, y-1] + b[x-1, y+1]]]];
a[n_] := b[n, 0][[2]];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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