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A247301 Rectangular array read upwards by columns: T = T(n,k) = number of paths from (0,0) to (n,k), where 0 >= k <= 2, consisting of segments given by the vectors (1,1), (2,1), (1,-1).

%I A247301 #13 Jan 24 2020 17:04:31
%S A247301 1,0,0,0,1,0,1,1,1,1,2,2,2,4,3,4,6,6,6,12,10,12,20,18,20,36,32,36,64,
%T A247301 56,64,112,100,112,200,176,200,352,312,352,624,552,624,1104,976,1104,
%U A247301 1952,1728,1952,3456,3056,3456,6112,5408,6112,10816,9568,10816
%N A247301 Rectangular array read upwards by columns: T = T(n,k) = number of paths from (0,0) to (n,k), where 0 >= k <= 2, consisting of segments given by the vectors (1,1), (2,1), (1,-1).
%H A247301 Clark Kimberling, <a href="/A247301/b247301.txt">Table of n, a(n) for n = 0..1000</a>
%F A247301 (row 0, the bottom row):  r(n) = 2*r(n-2) + 2*r(n-3), with r(0) = 1, r(1) = 0, r(2) = 1, r(3) = 1;
%F A247301 (row 1, the middle row):  r(n) = 2*r(n-2) + 2*r(n-3), with r(0) = 0, r(1) = 2, r(2) = 1, r(3) = 2;
%F A247301 (row 2, the top row):  r(n) = 2*r(n-2) + 2*r(n-3), with r(0) = 0, r(1) = 0, r(2) = 1, r(3) = 2.
%F A247301 From _Chai Wah Wu_, Jan 24 2020: (Start)
%F A247301 a(n) = 2*a(n-6) + 2*a(n-9) for n > 14.
%F A247301 G.f.: (-x^14 - 2*x^11 + x^9 - x^8 - x^7 + x^6 - x^4 - 1)/(2*x^9 + 2*x^6 - 1). (End)
%e A247301 First 10 columns:
%e A247301 0 .. 0 .. 1 .. 2 .. 3 .. 6 .. 10 .. 18 .. 32 .. 56
%e A247301 0 .. 1 .. 1 .. 2 .. 4 .. 6 .. 12 .. 20 .. 36 .. 64
%e A247301 1 .. 0 .. 1 .. 1 .. 2 .. 4 .. 6 ... 12 .. 20 .. 36
%e A247301 T(4,1) counts these 4 paths, given as vector sums applied to (0,0):
%e A247301 (1,1) + (1,-1) + (2,1);
%e A247301 (2,1) + (1,-1) + (1,-1);
%e A247301 (2,1) + (1,1) + (1,-1);
%e A247301 (1,1) + (2,1) + (1,-1).
%t A247301 t[0, 0] = 1; t[0, 1] = 0; t[0, 2] = 0;
%t A247301 t[1, 0] = 0; t[1, 1] = 1; t[1, 2] = 0;
%t A247301 t[2, 0] = 1; t[2, 1] = 1; t[2, 2] = 1; t[n_, 0] := t[n, 0] = t[n - 1, 1];
%t A247301 t[n_, 1] := t[n, 1] = t[n - 1, 0] + t[n - 1, 2] + t[n - 2, 0];
%t A247301 t[n_, 2] := t[n, 2] = t[n - 1, 1] + t[n - 2, 1];
%t A247301 TableForm[Reverse[Transpose[Table[t[n, k], {n, 0, 12}, {k, 0, 2}]]]]
%t A247301 Flatten[Table[t[n, k], {n, 0, 20}, {k, 0, 2}]]   (* A247301 *)
%Y A247301 Cf.  A247049, A247302.
%K A247301 nonn,tabf,easy
%O A247301 0,11
%A A247301 _Clark Kimberling_, Sep 11 2014