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A247354 Number of paths from (0,1) to (n,0), with vertices (i,k) satisfying 0 <= k <= 3, consisting of segments given by the vectors (1,1), (1,2), (1,-1).

%I A247354 #7 Sep 18 2014 05:17:03
%S A247354 0,1,0,2,2,5,10,17,38,66,138,257,508,981,1900,3702,7154,13925,26966,
%T A247354 52381,101594,197150,382578,742257,1440440,2794777,5423256,10522954,
%U A247354 20418882,39620597,76879298,149176601,289460206,561667802,1089854522,2114747217
%N A247354 Number of paths from (0,1) to (n,0), with vertices (i,k) satisfying 0 <= k <= 3, consisting of segments given by the vectors (1,1), (1,2), (1,-1).
%C A247354 Also, a(n) = number of strings s(0)..s(n) of integers such that s(0) = 1, s(n) = 0, and for i > 0,  s(i) is in {0,1,2,3} and s(i) - s(i-1) is in {-1,1,2} for 1 <= i <= n; also, a(n) = row 0 (the bottom row) of the array at A247352, and a(n+1) = row 1 of the same array.
%H A247354 Clark Kimberling, <a href="/A247354/b247354.txt">Table of n, a(n) for n = 0..1000</a>
%F A247354 Empirically, a(n) = 3*a(n-2) + 2*a(n-3) - a(n-4) and g.f. =  (x - x^3)/(1 - 3 x^2 - 2 x^3 + x^4).
%e A247354 a(5) counts these 5 paths, each represented by a vector sum applied to (0,1):
%e A247354 (1,1) + (1,1) + (1,-1) + (1,-1) + (1,-1)
%e A247354 (1,1) + (1,-1) + (1,1) + (1,-1) + (1,-1)
%e A247354 (1,-1) + (1,1) + (1,1) + (1,-1) + (1,-1)
%e A247354 (1,1) + (1,-1) + (1,-1) + (1,1) + (1,-1)
%e A247354 (1,-1) + (1,1) + (1,-1) + (1,1) + (1,-1)
%t A247354 z = 50; t[0, 0] = 0; t[0, 1] = 1; t[0, 2] = 0; t[0, 3] = 0;
%t A247354 t[1, 3] = 1; t[n_, 0] := t[n, 0] = t[n - 1, 1];
%t A247354 t[n_, 1] := t[n, 1] = t[n - 1, 0] + t[n - 1, 2]
%t A247354 t[n_, 2] := t[n, 2] = t[n - 1, 0] + t[n - 1, 1] + t[n - 1, 3]
%t A247354 t[n_, 3] := t[n, 3] = t[n - 1, 1] + t[n - 1, 2]
%t A247354 Table[t[n, 0], {n, 0, z}]   (* A247354*)
%Y A247354 Cf.  A247352, A247353, A247355.
%K A247354 nonn,easy
%O A247354 0,4
%A A247354 _Clark Kimberling_, Sep 15 2014