A247302 - OEIS

%I #12 Jan 24 2020 17:04:43

%S 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 64,72,128,112,128,224,200,224,400,352,400,704,624,704,1248,1104,1248,

%U 2208,1952,2208,3904,3456,3904,6912,6112,6912,12224,10816,12224

%N 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 Clark Kimberling, <a href="/A247302/b247302.txt">Table of n, a(n) for n = 0..1000</a>

%F (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 (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 (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 From _Chai Wah Wu_, Jan 24 2020: (Start)

%F a(n) = 2*a(n-6) + 2*a(n-9) for n > 8.

%F G.f.: (-x^8 - x^5 - x^3 - x)/(2*x^9 + 2*x^6 - 1). (End)

%e First 10 columns:

%e 0 .. 1 .. 1 .. 2 .. 4 .. 6 .. 12 .. 20 .. 36 .. 64

%e 1 .. 0 .. 2 .. 2 .. 4 .. 8 .. 12 .. 24 .. 40 .. 72

%e 0 .. 1 .. 0 .. 2 .. 2 .. 4 .. 8 ... 12 .. 24 .. 40

%e T(4,1) counts these 4 paths, given as vector sums applied to (0,0):

%e (1,1) + (1,-1) + (1,1) + (1,-1);

%e (1,1) + (1,-1) + (1,-1) + (1,1);

%e (1,-1) + (1,1) + (1,-1) + (1,1);

%e (1,-1) + (1,1) + (1,1) + (1,-1).

%t t[0, 0] = 0; t[0, 1] = 1; t[0, 2] = 0;

%t t[1, 0] = 1; t[1, 1] = 0; t[1, 2] = 1;

%t t[2, 0] = 0; t[2, 1] = 2; t[2, 2] = 1; 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[n - 2, 0];

%t t[n_, 2] := t[n, 2] = t[n - 1, 1] + t[n - 2, 1];

%t TableForm[Reverse[Transpose[Table[t[n, k], {n, 0, 12}, {k, 0, 2}]]]]

%t Flatten[Table[t[n, k], {n, 0, 20}, {k, 0, 2}]]   (* A247302 *)

%Y Cf.  A247050, A247301, A061275 (column sums).

%K nonn,tabf,easy

%O 0,8

%A _Clark Kimberling_, Sep 11 2014