This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A099331 #6 Aug 10 2016 00:53:43 %S A099331 0,0,0,2,1,4,3,12,16,40,56,122,197,408,695,1352,2368,4512,8096,15202, %T A099331 27529,51196,93339,172852,316368,584104,1071160,1974458,3625613, %U A099331 6677104,12269359,22583120,41513728,76387712,140454656,258398850,475182353 %N A099331 Number of Catalan knight paths from (0,0) to (n,3) in the region between and on the lines y=0 and y=3. (See A096587 for the definition of Catalan knight.). %F A099331 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 A099331 From _Chai Wah Wu_, Aug 09 2016: (Start) %F A099331 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 > 6. %F A099331 G.f.: x^3*(-x^2 + x - 2)/((-x^4 + 2*x^3 + 1)*(x^3 + x^2 + x - 1)). (End) %e A099331 a(6) counts 3 paths from (0,0) to (6,3); the final move in 1 %e A099331 path is from (4,2) and the final move in the other 2 paths %e A099331 is from (5,1). %Y A099331 Cf. A099328, A099329, A099330. %K A099331 nonn %O A099331 0,4 %A A099331 _Clark Kimberling_, Oct 12 2004