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A247302 Rectangular array read upwards by columns: T = T(n,k) = number of paths from (0,1) to (n,k), where 0 >= k <= 2, consisting of segments given by the vectors (1,1), (2,1), (1,-1).

%I A247302 #12 Jan 24 2020 17:04:43
%S A247302 0,1,0,1,0,1,0,2,1,2,2,2,2,4,4,4,8,6,8,12,12,12,24,20,24,40,36,40,72,
%T A247302 64,72,128,112,128,224,200,224,400,352,400,704,624,704,1248,1104,1248,
%U A247302 2208,1952,2208,3904,3456,3904,6912,6112,6912,12224,10816,12224
%N A247302 Rectangular array read upwards by columns: T = T(n,k) = number of paths from (0,1) to (n,k), where 0 >= k <= 2, consisting of segments given by the vectors (1,1), (2,1), (1,-1).
%H A247302 Clark Kimberling, <a href="/A247302/b247302.txt">Table of n, a(n) for n = 0..1000</a>
%F A247302 (row 0, the bottom row):  r(n) = 2*r(n-2) + 2*r(n-3), with r(0) = 0, r(1) = 1, r(2) = 0;
%F A247302 (row 1, the middle row):  r(n) = 2*r(n-2) + 2*r(n-3), with r(0) = 1, r(1) = 0, r(2) = 2;
%F A247302 (row 2, the top row):  r(n) = 2*r(n-2) + 2*r(n-3), with r(0) = 0, r(1) = 1, r(2) = 1.
%F A247302 From _Chai Wah Wu_, Jan 24 2020: (Start)
%F A247302 a(n) = 2*a(n-6) + 2*a(n-9) for n > 8.
%F A247302 G.f.: (-x^8 - x^5 - x^3 - x)/(2*x^9 + 2*x^6 - 1). (End)
%e A247302 First 10 columns:
%e A247302 0 .. 1 .. 1 .. 2 .. 4 .. 6 .. 12 .. 20 .. 36 .. 64
%e A247302 1 .. 0 .. 2 .. 2 .. 4 .. 8 .. 12 .. 24 .. 40 .. 72
%e A247302 0 .. 1 .. 0 .. 2 .. 2 .. 4 .. 8 ... 12 .. 24 .. 40
%e A247302 T(4,1) counts these 4 paths, given as vector sums applied to (0,0):
%e A247302 (1,1) + (1,-1) + (1,1) + (1,-1);
%e A247302 (1,1) + (1,-1) + (1,-1) + (1,1);
%e A247302 (1,-1) + (1,1) + (1,-1) + (1,1);
%e A247302 (1,-1) + (1,1) + (1,1) + (1,-1).
%t A247302 t[0, 0] = 0; t[0, 1] = 1; t[0, 2] = 0;
%t A247302 t[1, 0] = 1; t[1, 1] = 0; t[1, 2] = 1;
%t A247302 t[2, 0] = 0; t[2, 1] = 2; t[2, 2] = 1; t[n_, 0] := t[n, 0] = t[n - 1, 1];
%t A247302 t[n_, 1] := t[n, 1] = t[n - 1, 0] + t[n - 1, 2] + t[n - 2, 0];
%t A247302 t[n_, 2] := t[n, 2] = t[n - 1, 1] + t[n - 2, 1];
%t A247302 TableForm[Reverse[Transpose[Table[t[n, k], {n, 0, 12}, {k, 0, 2}]]]]
%t A247302 Flatten[Table[t[n, k], {n, 0, 20}, {k, 0, 2}]]   (* A247302 *)
%Y A247302 Cf.  A247050, A247301, A061275 (column sums).
%K A247302 nonn,tabf,easy
%O A247302 0,8
%A A247302 _Clark Kimberling_, Sep 11 2014