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A099331
Number of Catalan knight paths from (0,0) to (n,3) in the region between and on the lines y=0 and y=3. (See A096587 for the definition of Catalan knight.).
4
0, 0, 0, 2, 1, 4, 3, 12, 16, 40, 56, 122, 197, 408, 695, 1352, 2368, 4512, 8096, 15202, 27529, 51196, 93339, 172852, 316368, 584104, 1071160, 1974458, 3625613, 6677104, 12269359, 22583120, 41513728, 76387712, 140454656, 258398850, 475182353
OFFSET
0,4
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 > 6.
G.f.: x^3*(-x^2 + x - 2)/((-x^4 + 2*x^3 + 1)*(x^3 + x^2 + x - 1)). (End)
EXAMPLE
a(6) counts 3 paths from (0,0) to (6,3); the final move in 1
path is from (4,2) and the final move in the other 2 paths
is from (5,1).
CROSSREFS
KEYWORD
nonn
AUTHOR
Clark Kimberling, Oct 12 2004
STATUS
approved