cp's OEIS Frontend

A247355 Number of paths from (0,1) 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 A247355 #8 Sep 18 2014 05:16:29
%S A247355 0,1,2,3,8,12,28,49,100,191,370,724,1392,2721,5254,10223,19812,38456,
%T A247355 74628,144769,280984,545107,1057862,2052520,3982816,7728177,14995626,
%U A247355 29097643,56460416,109556004,212580908,412491201,800394316,1553079415,3013584442
%N A247355 Number of paths from (0,1) 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 A247355 Also, a(n) = number of strings s(0)..s(n) of integers such that s(0) = 1, 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 A247352.
%H A247355 Clark Kimberling, <a href="/A247355/b247355.txt">Table of n, a(n) for n = 0..1000</a>
%F A247355 Empirically, a(n) = 3*a(n-2) + 2*a(n-3) - a(n-4) and g.f. =  (x + 2 x^3)/(1 - 3 x^2 - 2 x^3 + x^4).
%e A247355 a(3) counts these 3 paths, each represented by a vector sum applied to (0,1):
%e A247355 (1,1) + (1,1) + (1,-1);
%e A247355 (1,1) + (1,-1) + (1,1);
%e A247355 (1,-1) + (1,1) + (1,1).
%t A247355 z = 50; t[0, 0] = 0; t[0, 1] = 1; t[0, 2] = 0; t[0, 3] = 0;
%t A247355 t[1, 3] = 1; t[n_, 0] := t[n, 0] = t[n - 1, 1];
%t A247355 t[n_, 1] := t[n, 1] = t[n - 1, 0] + t[n - 1, 2];
%t A247355 t[n_, 2] := t[n, 2] = t[n - 1, 0] + t[n - 1, 1] + t[n - 1, 3];
%t A247355 t[n_, 3] := t[n, 3] = t[n - 1, 1] + t[n - 1, 2];
%t A247355 Table[t[n, 2], {n, 0, z}]   (* A247355 *)
%Y A247355 Cf.  A247352, A247353, A247354.
%K A247355 nonn,easy
%O A247355 0,3
%A A247355 _Clark Kimberling_, Sep 15 2014