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A099329 Number of Catalan knight paths from (0,0) to (n,1) in the region between and on the lines y=0 and y=3. (See A096587 for the definition of Catalan knight.). 3
0, 0, 1, 1, 3, 2, 7, 10, 26, 38, 79, 127, 261, 452, 877, 1540, 2916, 5244, 9837, 17853, 33159, 60486, 111923, 204974, 378334, 694018, 1278939, 2348795, 4325129, 7948424, 14628953, 26893256, 49482888, 90987448, 167388697, 307825273 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

LINKS

Table of n, a(n) for n=1..36.

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

EXAMPLE

a(6) counts 7 paths from (0,0) to (6,1); the final move in 4 of the paths is from the point (5,3), the final move in 1 path is from (4,2) and the final move in the other 3 paths is from (4,0).

CROSSREFS

Cf. A099328, A099330, A099331.

Sequence in context: A034423 A193859 A174330 * A182871 A143329 A053440

Adjacent sequences:  A099326 A099327 A099328 * A099330 A099331 A099332

KEYWORD

nonn

AUTHOR

Clark Kimberling, Oct 12 2004

STATUS

approved

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Last modified September 27 13:48 EDT 2021. Contains 347688 sequences. (Running on oeis4.)