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%I #9 Aug 10 2016 00:53:26
%S 0,0,1,1,3,2,7,10,26,38,79,127,261,452,877,1540,2916,5244,9837,17853,
%T 33159,60486,111923,204974,378334,694018,1278939,2348795,4325129,
%U 7948424,14628953,26893256,49482888,90987448,167388697,307825273
%N 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.).
%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^3*(x^3 - x^2 - 1)/((-x^4 + 2*x^3 + 1)*(x^3 + x^2 + x - 1)). (End)
%e 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).
%Y Cf. A099328, A099330, A099331.
%K nonn
%O 1,5
%A _Clark Kimberling_, Oct 12 2004
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