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A247326 Number of paths from (0,0) to (n,3), with vertices (i,k) satisfying 0 <= k <= 3, consisting of segments given by the vectors (1,1), (1,2), (1,-1).

%I A247326 #8 Sep 16 2014 02:34:25
%S A247326 0,0,2,2,6,10,20,40,74,150,282,558,1072,2088,4050,7850,15254,29562,
%T A247326 57412,111344,216106,419294,813594,1578750,3063264,5944144,11533698,
%U A247326 22380210,43426118,84263882,163505076,317263672,615616874,1194537286,2317872890,4497581934
%N A247326 Number of paths from (0,0) to (n,3), with vertices (i,k) satisfying 0 <= k <= 3, consisting of segments given by the vectors (1,1), (1,2), (1,-1).
%C A247326 Also, a(n) = number of strings s(0)..s(n) of integers such that s(0) = 0, s(n) = 3, 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 3 of the array at A247321.
%H A247326 Clark Kimberling, <a href="/A247326/b247326.txt">Table of n, a(n) for n = 0..1000</a>
%F A247326 Empirically, a(n) = 3*a(n-2) + 2*a(n-3) - a(n-4) and g.f. =  (2*x^2 + x^3)/(1 - 3 x^2 - 2 x^3 + x^4).
%e A247326 a(4) counts these 6 paths, each represented by a vector sum applied to (0,0):
%e A247326 (1,2) + (1,1) + (1,-1) + (1,1);
%e A247326 (1,1) + (1,2) + (1,-1) + (1,1);
%e A247326 (1,2) + (1,-1) + (1,1) + (1,1);
%e A247326 (1,1) + (1,-1) + (1,2) + (1,1);
%e A247326 (1,1) + (1,-1) + (1,1) + (1,2);
%e A247326 (1,1) + (1,1) + (1,-1) + (1,2).
%t A247326 z = 25; t[0, 0] = 1; t[0, 1] = 0; t[0, 2] = 0; t[0, 3] = 0;
%t A247326 t[1, 3] = 0; t[n_, 0] := t[n, 0] = t[n - 1, 1];
%t A247326 t[n_, 1] := t[n, 1] = t[n - 1, 0] + t[n - 1, 2];
%t A247326 t[n_, 2] := t[n, 2] = t[n - 1, 0] + t[n - 1, 1] + t[n - 1, 3];
%t A247326 t[n_, 3] := t[n, 3] = t[n - 1, 1] + t[n - 1, 2];
%t A247326 Table[t[n, 3], {n, 0, z}];  (* A247326 *)
%Y A247326 Cf.  A247049, A247321, A247322, A247325.
%K A247326 nonn,easy
%O A247326 0,3
%A A247326 _Clark Kimberling_, Sep 13 2014