%I #9 Sep 16 2014 02:34:51
%S 1,0,1,1,2,5,7,18,29,63,116,229,445,856,1677,3229,6298,12185,23675,
%T 45922,89097,172931,335460,651065,1263145,2451184,4756105,9228777,
%U 17907538,34747357,67424063,130828370,253859365,492585879,955810772,1854647997,3598744709
%N Number of paths from (0,0) 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 Also, a(n) = number of strings s(0)..s(n) of integers such that s(0) = 0, s(n) = 0, and for i > 0, s(i) is in {0,1,2,3} and s(i)  s(i1) is in {1,1,2} for 1 <= i <= n; also, a(n) = row 0 (the bottom row) of the array at A247321, and a(n+1) = row 1 of the same array.
%H Clark Kimberling, <a href="/A247323/b247323.txt">Table of n, a(n) for n = 0..1000</a>
%F Empirically, a(n) = 3*a(n2) + 2*a(n3)  a(n4) and g.f. = (1 + 2*x^2  x^3)/(1  3 x^2  2 x^3 + x^4).
%e a(5) counts these 5 paths, each represented by a vector sum applied to (0,0):
%e (1,2) + (1,1) + (1,1) + (1,1) + (1,1);
%e (1,1) + (1,2) + (1,1) + (1,1) + (1,1);
%e (1,2) + (1,1) + (1,1) + (1,1) + (1,1);
%e (1,1) + (1,1) + (1,2) + (1,1) + (1,1);
%e (1,2) + (1,1) + (1,1) + (1,1) + (1,1).
%t z = 25; t[0, 0] = 1; t[0, 1] = 0; t[0, 2] = 0; t[0, 3] = 0;
%t t[1, 3] = 0; t[n_, 0] := t[n, 0] = t[n  1, 1];
%t t[n_, 1] := t[n, 1] = t[n  1, 0] + t[n  1, 2];
%t t[n_, 2] := t[n, 2] = t[n  1, 0] + t[n  1, 1] + t[n  1, 3];
%t t[n_, 3] := t[n, 3] = t[n  1, 1] + t[n  1, 2];
%t Table[t[n, 0], {n, 0, z}]; (* A247323 *)
%Y Cf. A247049, A247321, A247322, A247325, A247326.
%K nonn,easy
%O 0,5
%A _Clark Kimberling_, Sep 13 2014
