A099328 Number of Catalan knight paths from (0,0) to (n,0) in the region between and on the lines y=0 and y=3. (See A096587 for the definition of Catalan knight.).

1, 0, 1, 0, 2, 2, 8, 8, 21, 28, 69, 108, 226, 370, 736, 1280, 2473, 4392, 8281, 14920, 27874, 50706, 94088, 171880, 317693, 582116, 1073853, 1970836, 3630914, 6669730, 12279296, 22568896, 41533777, 76360464, 140493041, 258344528, 475256898

Offset

1,5

Links

Table of n, a(n) for n=1..37.

Index entries for linear recurrences with constant coefficients, signature (1,1,-1,3,1,1,-1).

Formula

Taking A099328 to A099331 as the rows of an array T, the recurrences for these row sequences are given for n>=2 by T(n, 0) = T(n-1, 2) + T(n-2, 1), T(n, 1) = T(n-1, 3) + T(n-2, 0) + T(n-2, 2), T(n, 2) = T(n-1, 0) + T(n-2, 1) + T(n-2, 3), T(n, 3) = T(n-1, 1) + T(n-2, 2), with initial values T(0, 0)=1, T(1, 2)=1.

From Chai Wah Wu, Aug 09 2016: (Start)

a(n) = a(n-1) + a(n-2) - a(n-3) + 3*a(n-4) + a(n-5) + a(n-6) - a(n-7) for n > 7.

G.f.: x*(1 - x - 2*x^4)/((x^4 - 2*x^3 - 1)*(x^3 + x^2 + x - 1)). (End)

Example

a(6) counts 8 paths from (0,0) to (6,0); the final move in 5 of the paths is from the point (5,2) and the final move in the other 3 paths is from (4,1).

Mathematica

LinearRecurrence[{1, 1, -1, 3, 1, 1, -1}, {1, 0, 1, 0, 2, 2, 8}, 40] (* Harvey P. Dale, Aug 11 2017 *)

Crossrefs

Cf. A099329, A099330, A099331.

Sequence in context: A060818 A082887 A137583 * A073090 A120544 A155950

Adjacent sequences: A099325 A099326 A099327 * A099329 A099330 A099331

Keyword

nonn

Author

Clark Kimberling, Oct 12 2004

Status

approved