A247323 -id:A247323 - cp's OEIS frontend

A247321 Rectangular array read upwards by columns: T = T(n,k) = number of paths from (0,0) to (n,k), where 0 <= k <= 3, consisting of segments given by the vectors (1,1), (1,2), (1,-1).

1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 2, 1, 2, 4, 2, 2, 5, 5, 6, 5, 7, 13, 10, 7, 18, 22, 20, 18, 29, 45, 40, 29, 63, 87, 74, 63, 116, 166, 150, 116, 229, 329, 282, 229, 445, 627, 558, 445, 856, 1232, 1072, 856, 1677, 2373, 2088, 1677, 3229, 4621, 4050, 3229

Offset: 0

Clark Kimberling, Sep 13 2014

Also, T(n,k) = number of strings s(0)..s(n) of integers such that s(0) = 0, s(n) = k, 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.

			First 10 columns:
0 .. 0 .. 2 .. 2 .. 6 .. 10 .. 20 .. 40 .. 74 .. 150
0 .. 1 .. 1 .. 4 .. 5 .. 13 .. 22 .. 45 .. 87 .. 166
0 .. 1 .. 1 .. 2 .. 5 .. 7 ... 18 .. 29 .. 63 .. 116
1 .. 0 .. 1 .. 1 .. 2 .. 5 ... 7 ... 18 .. 29 .. 63
T(3,2) counts these 4 paths, given as vector sums applied to (0,0):
(1,2) + (1,1) + (1, -1)
(1,1) + (1,2) + (1,-1)
(1,2) + (1,-1) + (1,1)
(1,1) + (1,-1) + (1,2)
Partial sums of second components in each vector sum give the 3 integer strings described in Comments:  (0,2,3,2), (0,1,3,2), (0,2,1,2), (0,1,0,2).

Clark Kimberling, Table of n, a(n) for n = 0..1000

Cf. A247049, A247322, A247323, A247325, A247326.

z = 25; t[0, 0] = 1; t[0, 1] = 0; t[0, 2] = 0; t[0, 3] = 0;
t[1, 3] = 0; t[n_, 0] := t[n, 0] = t[n - 1, 1];
t[n_, 1] := t[n, 1] = t[n - 1, 0] + t[n - 1, 2];
t[n_, 2] := t[n, 2] = t[n - 1, 0] + t[n - 1, 1] + t[n - 1, 3];
t[n_, 3] := t[n, 3] = t[n - 1, 1] + t[n - 1, 2];
u = Flatten[Table[t[n, k], {n, 0, z}, {k, 0, 3}]] (* A247321 *)
TableForm[Reverse[Transpose[Table[t[n, k], {n, 0, 12}, {k, 0, 3}]]]]
u1 = Table[t[n, k], {n, 0, z}, {k, 0, 3}];
v = Map[Total, u1]  (* A247322 column sums *)
Table[t[n, 0], {n, 0, z}]   (* A247323, row 0 *)
Table[t[n, 1], {n, 0, z}]   (* A247323 shifted, row 1 *)
Table[t[n, 2], {n, 0, z}]   (* A247325, row 2 *)
Table[t[n, 3], {n, 0, z}]   (* A247326, row 3 *)

The four rows and the column sums all empirically satisfy the linear recurrence r(n) = 3*r(n-2) + 2*r(n-3) - r(n-4), with g.f. of the form p(x)/q(x), where q(x) = 1 - 3 x^2 - 2 x^3 + x^4.  Initial terms and p(x) follow:
(row 0, the bottom row):  1,0,1,1; 1 - 2*x^2 - x^3
(row 1):  0,1,1,2; x + x^2 - x^3
(row 2):  0,1,1,4; x + x^2 + x^3
(row 3):  0,0,1,1; 2x^2 + 2x^3
(n-th column sum) = 1,2,5,9; 1 + 2*x + 2*x^2 + x^3.

A247322 Number of paths from (0,0) to the line x = n, each consisting of segments given by the vectors (1,1), (1,2), (1,-1), with vertices (i,k) satisfying 0 <= k <= 3.

Original entry on oeis.org

1, 2, 5, 9, 18, 35, 67, 132, 253, 495, 956, 1859, 3605, 6994, 13577, 26333, 51114, 99159, 192431, 373372, 724497, 1405819, 2727804, 5293079, 10270553, 19929026, 38670013, 75035105, 145597538, 282516315, 548192811, 1063708916, 2064013525, 4004996055

Offset: 0

Clark Kimberling, Sep 13 2014

Also, a(n) = number of strings s(0)..s(n) of integers such that s(0) = 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) = n-th column sum of the array at A247321.

			a(2) counts these 5 paths, each represented by a vector sum applied to (0,0): (0,2) + (0,1); (0,1) + (0,2); (0,1) + (0,1); (0,2) + (0,-1), (0,1) + (0,-1).

Clark Kimberling, Table of n, a(n) for n = 0..1000

Cf. A247049, A247321, A247323, A247325, A247326.

z = 25; t[0, 0] = 1; t[0, 1] = 0; t[0, 2] = 0; t[0, 3] = 0;
t[1, 3] = 0; t[n_, 0] := t[n, 0] = t[n - 1, 1];
t[n_, 1] := t[n, 1] = t[n - 1, 0] + t[n - 1, 2];
t[n_, 2] := t[n, 2] = t[n - 1, 0] + t[n - 1, 1] + t[n - 1, 3];
t[n_, 3] := t[n, 3] = t[n - 1, 1] + t[n - 1, 2];
u = Flatten[Table[t[n, k], {n, 0, z}, {k, 0, 3}]] (* A247321 *)
TableForm[Reverse[Transpose[Table[t[n, k], {n, 0, 12}, {k, 0, 3}]]]]
u1 = Table[t[n, k], {n, 0, z}, {k, 0, 3}];
v = Map[Total, u1]  (* A247322 column sums *)

A247322(n) = A247323(n) + A247323(n+1) + A247325(n) + A247326(n).

Empirically, a(n) = 3*a(n-2) + 2*a(n-3) - a(n-4) and g.f. = (1 + 2*x + 2*x^2 + x^3)/(1 - 3 x^2 - 2 x^3 + x^4).

cp's OEIS Frontend

A247321 Rectangular array read upwards by columns: T = T(n,k) = number of paths from (0,0) to (n,k), where 0 <= k <= 3, consisting of segments given by the vectors (1,1), (1,2), (1,-1).

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