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A099330
Number of Catalan knight paths from (0,0) to (n,2) in the region between and on the lines y=0 and y=3. (See A096587 for the definition of Catalan knight.).
3
0, 1, 0, 1, 1, 5, 6, 14, 18, 43, 70, 147, 243, 475, 828, 1596, 2852, 5365, 9676, 18037, 32853, 60929, 111394, 205770, 377142, 695519, 1276818, 2351975, 4320935, 7954167, 14620472, 26904824, 49467208, 91010153, 167357080, 307868201
OFFSET
1,6
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^2*(x^3 - x + 1)/((-x^4 + 2*x^3 + 1)*(x^3 + x^2 + x - 1)). (End)
EXAMPLE
a(6) counts 6 paths from (0,0) to (6,2); the final move in 1 path is from the point (4,3), the final move in 3 paths is from (4,1) and the final move in the other 2 paths is from (5,0).
CROSSREFS
KEYWORD
nonn
AUTHOR
Clark Kimberling, Oct 12 2004
STATUS
approved