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A127619
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Number of walks from (0,0) to (n,n) in the region 0 <= x-y <= 5 with the steps (1,0), (0, 1), (2,0) and (0,2).
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3
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1, 1, 5, 22, 117, 654, 3674, 20763, 117349, 663529, 3751874, 21215245, 119963514, 678345474, 3835772387, 21689760681, 122646936325, 693519457822, 3921575652821, 22174944672838, 125390459051898, 709032985366923
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OFFSET
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0,3
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LINKS
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Table of n, a(n) for n=0..21.
Arvind Ayyer and Doron Zeilberger, The Number of [Old-Time] Basketball games with Final Score n:n where the Home Team was never losing but also never ahead by more than w Points, arXiv:math/0610734 [math.CO], 2006-2007.
Index entries for linear recurrences with constant coefficients, signature (5, 6, -11, -12, 4).
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FORMULA
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G.f.: (1-4x-6x^2+2x^3)/(1-5x-6x^2+11x^3+12x^4-4x^5). [Typo corrected by Jean-François Alcover, Dec 10 2018]
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EXAMPLE
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a(2)=5 because we can reach (2,2) in the following ways:
(0,0),(1,0),(1,1),(2,1),(2,2)
(0,0),(2,0),(2,2)
(0,0),(1,0),(2,0),(2,2)
(0,0),(2,0),(2,1),(2,2)
(0,0),(1,0),(2,0),(2,1),(2,2)
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MATHEMATICA
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LinearRecurrence[{5, 6, -11, -12, 4}, {1, 1, 5, 22, 117}, 22] (* Jean-François Alcover, Dec 10 2018 *)
b[n_, k_] := Boole[n >= 0 && k >= 0 && 0 <= n - k <= 5];
T[0, 0] = T[1, 1] = 1; T[n_, k_] /; b[n, k] == 1 := T[n, k] = b[n-2, k]* T[n-2, k] + b[n-1, k]*T[n-1, k] + b[n, k-2]*T[n, k-2] + b[n, k-1]*T[n, k-1]; T[_, _] = 0;
a[n_] := T[n, n];
Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Apr 03 2019 *)
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CROSSREFS
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Cf. A000108, A046717, A122951, A127617, A127618, A127620.
Sequence in context: A355398 A005033 A127618 * A127620 A122951 A331836
Adjacent sequences: A127616 A127617 A127618 * A127620 A127621 A127622
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KEYWORD
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nonn,easy,walk
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AUTHOR
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Arvind Ayyer, Jan 20 2007
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STATUS
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approved
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