

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
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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(n1, 2) + T(n2, 1), T(n, 1) = T(n1, 3) + T(n2, 0) + T(n2, 2), T(n, 2) = T(n1, 0) + T(n2, 1) + T(n2, 3), T(n, 3) = T(n1, 1) + T(n2, 2), with initial values T(0, 0)=1, T(1, 2)=1.
From Chai Wah Wu, Aug 09 2016: (Start)
a(n) = a(n1) + a(n2)  a(n3) + 3*a(n4) + a(n5) + a(n6)  a(n7) 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



