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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.). 4
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified May 19 13:24 EDT 2022. Contains 353833 sequences. (Running on oeis4.)