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A247323 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).

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