A099330 - OEIS

%I #10 Aug 10 2016 00:53:33

%S 0,1,0,1,1,5,6,14,18,43,70,147,243,475,828,1596,2852,5365,9676,18037,

%T 32853,60929,111394,205770,377142,695519,1276818,2351975,4320935,

%U 7954167,14620472,26904824,49467208,91010153,167357080,307868201

%N 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.).

%F 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.

%F From _Chai Wah Wu_, Aug 09 2016: (Start)

%F 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.

%F G.f.: -x^2*(x^3 - x + 1)/((-x^4 + 2*x^3 + 1)*(x^3 + x^2 + x - 1)). (End)

%e 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).

%Y Cf. A099328, A099329, A099331.

%K nonn

%O 1,6

%A _Clark Kimberling_, Oct 12 2004