cp's OEIS Frontend

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.).

%I A099328 #11 Nov 22 2024 05:35:15
%S A099328 1,0,1,0,2,2,8,8,21,28,69,108,226,370,736,1280,2473,4392,8281,14920,
%T A099328 27874,50706,94088,171880,317693,582116,1073853,1970836,3630914,
%U A099328 6669730,12279296,22568896,41533777,76360464,140493041,258344528,475256898
%N 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.).
%H A099328 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1,3,1,1,-1).
%F A099328 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 A099328 From _Chai Wah Wu_, Aug 09 2016: (Start)
%F A099328 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 A099328 G.f.: x*(1 - x - 2*x^4)/((x^4 - 2*x^3 - 1)*(x^3 + x^2 + x - 1)). (End)
%F A099328 2*a(n) = A001590(n)-(-1)^n*( A052922(n-1)+A052922(n-3)) . - _R. J. Mathar_, Nov 22 2024
%e A099328 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).
%t A099328 LinearRecurrence[{1,1,-1,3,1,1,-1},{1,0,1,0,2,2,8},40] (* _Harvey P. Dale_, Aug 11 2017 *)
%Y A099328 Cf. A099329, A099330, A099331.
%K A099328 nonn,easy
%O A099328 1,5
%A A099328 _Clark Kimberling_, Oct 12 2004