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A247353 Number of paths from (0,1) to the line x = n, each consisting of segments given by the vectors (1,1), (1,2), (1,-1), with vertices (i,k) satisfying 0 <= k <= 3.

%I A247353 #7 Sep 18 2014 05:17:39
%S A247353 1,3,5,11,20,40,77,149,291,561,1094,2116,4113,7975,15477,30035,58268,
%T A247353 113084,219397,425753,826091,1602969,3110382,6035336,11710993,
%U A247353 22723803,44093269,85558059,166016420,322136912,625072109,1212885517,2353473731,4566663857
%N A247353 Number of paths from (0,1) to the line x = n, each consisting of segments given by the vectors (1,1), (1,2), (1,-1), with vertices (i,k) satisfying 0 <= k <= 3.
%C A247353 Also, a(n) = number of strings s(0)..s(n) of integers such that s(0) = 1, 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) = n-th column sum of the array at A247352.
%H A247353 Clark Kimberling, <a href="/A247353/b247353.txt">Table of n, a(n) for n = 0..1000</a>
%F A247353 A247353(n) = A247354(n) + A247354(n+1) + A247355(n) + A247321(n).
%F A247353 Empirically, a(n) = 3*a(n-2) + 2*a(n-3) - a(n-4) and g.f. =  (1 + 3*x + 2*x^2)/(1 - 3 x^2 - 2 x^3 + x^4).
%e A247353 a(2) counts these 5 paths, each represented by a vector sum applied to (0,1):
%e A247353 (1,1) + (1,1) = (1,2) + (1,-1) = (1,-1) + (1,2) = (1,1) + (1,-1) = (1,-1) + (1,1).
%t A247353 z = 50; t[0, 0] = 0; t[0, 1] = 1; t[0, 2] = 0; t[0, 3] = 0;
%t A247353 t[1, 3] = 1; t[n_, 0] := t[n, 0] = t[n - 1, 1];
%t A247353 t[n_, 1] := t[n, 1] = t[n - 1, 0] + t[n - 1, 2]
%t A247353 t[n_, 2] := t[n, 2] = t[n - 1, 0] + t[n - 1, 1] + t[n - 1, 3]
%t A247353 t[n_, 3] := t[n, 3] = t[n - 1, 1] + t[n - 1, 2]
%t A247353 u = Flatten[Table[t[n, k], {n, 0, z}, {k, 0, 3}]] (* A247352 *)
%t A247353 u1 = Table[t[n, k], {n, 0, z}, {k, 0, 3}];
%t A247353 v = Map[Total, u1]  (* A247353 *)
%Y A247353 Cf.  A247322, A247352, A247354, A247355.
%K A247353 nonn,easy
%O A247353 0,2
%A A247353 _Clark Kimberling_, Sep 15 2014