A099328 Number of Catalan knight paths from (0,0) to (n,0) in the region between and on the lines y=0 and y=3. (See A096587 for the definition of Catalan knight.).
1, 0, 1, 0, 2, 2, 8, 8, 21, 28, 69, 108, 226, 370, 736, 1280, 2473, 4392, 8281, 14920, 27874, 50706, 94088, 171880, 317693, 582116, 1073853, 1970836, 3630914, 6669730, 12279296, 22568896, 41533777, 76360464, 140493041, 258344528, 475256898
Offset: 1
Examples
a(6) counts 8 paths from (0,0) to (6,0); the final move in 5 of the paths is from the point (5,2) and the final move in the other 3 paths is from (4,1).
Links
- Index entries for linear recurrences with constant coefficients, signature (1,1,-1,3,1,1,-1).
Programs
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Mathematica
LinearRecurrence[{1,1,-1,3,1,1,-1},{1,0,1,0,2,2,8},40] (* Harvey P. Dale, Aug 11 2017 *)
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*(1 - x - 2*x^4)/((x^4 - 2*x^3 - 1)*(x^3 + x^2 + x - 1)). (End)