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A190586
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Number of two-sided n-step prudent walks ending on the top side of their box, avoiding two or more consecutive west steps and south steps.
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2
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1, 3, 6, 15, 35, 83, 195, 460, 1085, 2560, 6039, 14247, 33613, 79306, 187114, 441477, 1041626, 2457630, 5798569, 13681202, 32279488, 76160166, 179691649, 423961718, 1000285928, 2360046161, 5568211498, 13137414580, 30995819288, 73129978187, 172538870438
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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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Recurrence: (n+1)*(7*n^2 - 194*n + 926)*a(n) = n*(42*n^2 - 1143*n + 5320)*a(n-1) - (84*n^3 - 2349*n^2 + 13043*n - 13734)*a(n-2) + (77*n^3 - 2267*n^2 + 15130*n - 27824)*a(n-3) - (63*n^3 - 1893*n^2 + 13180*n - 24672)*a(n-4) - (7*n^3 - 313*n^2 + 3298*n - 8548)*a(n-5) + (77*n^3 - 2309*n^2 + 16800*n - 34964)*a(n-6) + 2*(14*n^3 - 563*n^2 + 6962*n - 24347)*a(n-7) + (49*n^3 - 1456*n^2 + 10086*n - 21316)*a(n-8) - (7*n^3 - 208*n^2 + 1687*n - 2950)*a(n-9) - 6*(7*n^3 - 236*n^2 + 2187*n - 5968)*a(n-10) - 2*(7*n^3 - 250*n^2 + 2526*n - 7744)*a(n-11) + 2*(n-10)*(7*n^2 - 180*n + 739)*a(n-12). - Vaclav Kotesovec, Sep 03 2014
G.f.: 1/(2*t*(1-2*t-t^2+t^3)*(1-2*t-2*t^3))*((1-2*t)*(1-t)*sqrt((1-t-t^3)^2-4*t^4)-(1+t)*(1-7*t+14*t^2-11*t^3+10*t^4-4*t^5)), see sequence 11 in link. - Michel Marcus, May 06 2015
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EXAMPLE
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a(2) = 6 since there are 6 such walks: WN, NW, EN, NE, EE, NN.
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MAPLE
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b:= proc(d, i, n, x, y) option remember;
`if`(n=0, `if`(y=0, 1, 0),
`if`(d<>3, b(1, x=0, n-1, max(x-1, 0), y), 0) +
`if`(d<>4, b(2, y=0, n-1, x, max(y-1, 0)), 0) +
`if`(d=0 or d=2 and i, b(3, false, n-1, x+1, y), 0) +
`if`(d=0 or d=1 and i, b(4, false, n-1, x, y+1), 0))
end:
a:= n-> b(0, false, n, 0, 0):
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MATHEMATICA
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CoefficientList[Series[1/(2 x (1-2 x - x^2 + x^3) (1 - 2 x - 2 x^3)) ((1 - 2 x) (1 - x) Sqrt[(1 - x - x^3)^2 - 4 x^4] - (1 + x) (1 - 7 x + 14 x^2 - 11 x^3 + 10 x^4 - 4 x^5)), {x, 0, 50}], x] (* Vincenzo Librandi, May 07 2015 *)
b[d_, i_, n_, x_, y_] := b[d, i, n, x, y] = If[n == 0, If[y == 0, 1, 0], If[d != 3, b[1, x == 0, n-1, Max[x-1, 0], y], 0] + If[d != 4, b[2, y == 0, n-1, x, Max[y-1, 0]], 0] + If[d == 0 || d == 2 && i, b[3, False, n-1, x+1, y], 0] + If[d == 0 || d == 1 && i, b[4, False, n-1, x, y+1], 0]]; a[n_] := b[0, False, n, 0, 0]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Oct 24 2016, after Alois P. Heinz *)
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CROSSREFS
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KEYWORD
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nonn,walk
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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