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A247325 Number of paths from (0,0) to (n,2), with vertices (i,k) satisfying 0 <= k <= 3, consisting of segments given by the vectors (1,1), (1,2), (1,-1).

%I A247325 #10 Sep 16 2014 02:34:37
%S A247325 0,1,1,4,5,13,22,45,87,166,329,627,1232,2373,4621,8956,17377,33737,
%T A247325 65422,127009,246363,478134,927685,1800119,3492960,6777593,13151433,
%U A247325 25518580,49516525,96081013,186435302,361757509,701951407,1362062118,2642933937,5128331659
%N A247325 Number of paths from (0,0) to (n,2), with vertices (i,k) satisfying 0 <= k <= 3, consisting of segments given by the vectors (1,1), (1,2), (1,-1).
%C A247325 Also, a(n) = number of strings s(0)..s(n) of integers such that s(0) = 0, s(n) = 2, 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 2 of the array at A247321.
%H A247325 Clark Kimberling, <a href="/A247325/b247325.txt">Table of n, a(n) for n = 0..1000</a>
%F A247325 Empirically, a(n) = 3*a(n-2) + 2*a(n-3) - a(n-4) and g.f. =  (x + x^2 + x^3)/(1 - 3 x^2 - 2 x^3 + x^4).
%e A247325 a(4) counts these 4 paths, each represented by a vector sum applied to (0,0):
%e A247325 (1,2) + (1,1) + (1,-1);
%e A247325 (1,1) + (1,2) + (1,-1);
%e A247325 (1,2) + (1,-1) + (1,1);
%e A247325 (1,1) + (1,-1) + (1,2).
%t A247325 z = 25; t[0, 0] = 1; t[0, 1] = 0; t[0, 2] = 0; t[0, 3] = 0;
%t A247325 t[1, 3] = 0; t[n_, 0] := t[n, 0] = t[n - 1, 1];
%t A247325 t[n_, 1] := t[n, 1] = t[n - 1, 0] + t[n - 1, 2];
%t A247325 t[n_, 2] := t[n, 2] = t[n - 1, 0] + t[n - 1, 1] + t[n - 1, 3];
%t A247325 t[n_, 3] := t[n, 3] = t[n - 1, 1] + t[n - 1, 2];
%t A247325 Table[t[n, 2], {n, 0, z}];  (* A247325 *)
%Y A247325 Cf.  A247049, A247321, A247322, A247326.
%K A247325 nonn,easy
%O A247325 0,4
%A A247325 _Clark Kimberling_, Sep 13 2014