A127619 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).

1, 1, 5, 22, 117, 654, 3674, 20763, 117349, 663529, 3751874, 21215245, 119963514, 678345474, 3835772387, 21689760681, 122646936325, 693519457822, 3921575652821, 22174944672838, 125390459051898, 709032985366923

Offset: 0

Author

Arvind Ayyer, Jan 20 2007

Keywords

• nonn
• easy
• walk

Examples

			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)
		

Links

• 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).

Crossrefs

Cf. A000108, A046717, A122951, A127617, A127618, A127620.

Programs

Mathematica

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 *)

Formula

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]