A247049 -id:A247049 - 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).

A247325 Number of paths from (0,0) to (n,2), with vertices (i,k) satisfying 0 <= k <= 3, consisting of segments given by the vectors (1,1), (1,2), (1,-1).

Original entry on oeis.org

0, 1, 1, 4, 5, 13, 22, 45, 87, 166, 329, 627, 1232, 2373, 4621, 8956, 17377, 33737, 65422, 127009, 246363, 478134, 927685, 1800119, 3492960, 6777593, 13151433, 25518580, 49516525, 96081013, 186435302, 361757509, 701951407, 1362062118, 2642933937, 5128331659

Offset: 0

Clark Kimberling, Sep 13 2014

Also, a(n) = number of strings s(0)..s(n) of integers such that s(0) = 0, 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 A247321.

			a(4) counts these 4 paths, each represented by a vector sum 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).

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

Cf. A247049, A247321, A247322, 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];
Table[t[n, 2], {n, 0, z}];  (* A247325 *)

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

A247326 Number of paths from (0,0) to (n,3), with vertices (i,k) satisfying 0 <= k <= 3, consisting of segments given by the vectors (1,1), (1,2), (1,-1).

Original entry on oeis.org

0, 0, 2, 2, 6, 10, 20, 40, 74, 150, 282, 558, 1072, 2088, 4050, 7850, 15254, 29562, 57412, 111344, 216106, 419294, 813594, 1578750, 3063264, 5944144, 11533698, 22380210, 43426118, 84263882, 163505076, 317263672, 615616874, 1194537286, 2317872890, 4497581934

Offset: 0

Clark Kimberling, Sep 13 2014

Also, a(n) = number of strings s(0)..s(n) of integers such that s(0) = 0, s(n) = 3, 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 3 of the array at A247321.

			a(4) counts these 6 paths, each represented by a vector sum applied to (0,0):
(1,2) + (1,1) + (1,-1) + (1,1);
(1,1) + (1,2) + (1,-1) + (1,1);
(1,2) + (1,-1) + (1,1) + (1,1);
(1,1) + (1,-1) + (1,2) + (1,1);
(1,1) + (1,-1) + (1,1) + (1,2);
(1,1) + (1,1) + (1,-1) + (1,2).

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

Cf. A247049, A247321, A247322, A247325.

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];
Table[t[n, 3], {n, 0, z}];  (* A247326 *)

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

A247050 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), (1,2), (1,-1).

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 0, 2, 1, 2, 1, 2, 1, 4, 3, 4, 4, 5, 4, 9, 8, 9, 12, 13, 12, 22, 21, 22, 33, 34, 33, 56, 55, 56, 88, 89, 88, 145, 144, 145, 232, 233, 232, 378, 377, 378, 609, 610, 609, 988, 987, 988, 1596, 1597, 1596, 2585, 2584, 2585, 4180, 4181, 4180

Offset: 0

Clark Kimberling, Sep 11 2014

Also, T(n,k) = number of strings s(0)..s(n) of integers such that s(0) = 1, s(n) = k, and if i > 0, then s(i) is in {0,1,2} and s(i) - s(i-1) is in {1,2,-1}. The column sums form the Fibonacci sequence (A000045).

This is a 3-rowed array read upwards by columns. - N. J. A. Sloane, Sep 14 2014

			First 10 columns:
0 .. 1 .. 1 .. 2 .. 3 .. 5 .. 8 .. 13 .. 21 .. 34
1 .. 0 .. 2 .. 1 .. 4 .. 4 .. 9 .. 12 .. 22 .. 33
0 .. 1 .. 0 .. 2 .. 1 .. 4 .. 4 .. 9 ... 12 .. 22
T(4,1) counts these 4 paths, given as vector sums applied to (0,1):
(1,1) + (1,-1) + (1,1) + (1,-1);
(1,-1) + (1,1) + (1,1) + (1,-1);
(1,1) + (1,-1) + (1,-1) + (1,1);
(1,-1) + (1,1) + (1,-1) + (1,1)
Partial sums of second components in each vector sum give the 4 integer strings described in Comments: (1,2,1,2,1), (1,0,1,2,1), (1,2,1,0,1), (1,0,1,0,1).

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

Cf. A000045, A247049, A247050.

t[0, 0] = 0; t[0, 1] = 1; t[0, 2] = 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]; TableForm[ Reverse[Transpose[Table[t[n, k], {n, 0, 12}, {k, 0, 2}]]]] (* array *)
u = Flatten[Table[t[n, k], {n, 0, 20}, {k, 0, 2}]] (* sequence *)

Let F = A000045, the Fibonacci numbers.  Then (row 0, the bottom row) = F(n-1) - (-1)^n for n >= 0; (row 1, the middle row) = F(n) + (-1)^n for n >=0; (row 2, the top row) = F.
Conjectures from Chai Wah Wu, Apr 16 2025: (Start)
a(n) = 2*a(n-6) + a(n-9) for n > 8.
G.f.: (-x^8 - x^5 - x^3 - x)/(x^9 + 2*x^6 - 1). (End)

A247309 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), (1,0), (1,-1), (1,-2).

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 2, 3, 3, 5, 8, 8, 13, 21, 21, 34, 55, 55, 89, 144, 144, 233, 377, 377, 610, 987, 987, 1597, 2584, 2584, 4181, 6765, 6765, 10946, 17711, 17711, 28657, 46368, 46368, 75025, 121393, 121393, 196418, 317811, 317811, 514229, 832040, 832040

Offset: 0

Clark Kimberling, Sep 12 2014

Every member of T is a Fibonacci number, and the sum of the numbers in column n is A000045(2n+2).

			First 10 columns:
0 .. 1 .. 3 .. 8 .. 21 .. 55 .. 144 .. 377 .. 987 ... 2584
0 .. 1 .. 3 .. 8 .. 21 .. 55 .. 144 .. 377 .. 987 ... 2584
1 .. 1 .. 2 .. 5 .. 13 .. 34 .. 89 ... 233 .. 610 ... 1597
T(2,2) counts these 3 paths, given as vector sums applied to (0,0):
(1,2) + (1,0); (1,1) + (1,1); (1,0) + (1,2).

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

Cf. A247049, A247310, A000045.

t[0, 0] = 1; t[0, 1] = 0; t[0, 2] = 0; t[1, 2] = 1;
t[n_, 0] := t[n, 0] = t[n - 1, 0] + t[n - 1, 1];
t[n_, 1] := t[n, 1] = t[n - 1, 0] + t[n - 1, 1] + t[n - 1, 2];
t[n_, 2] := t[n, 2] = t[n - 1, 0] + t[n - 1, 1] + t[n - 1, 2]
TableForm[Reverse[Transpose[Table[t[n, k], {n, 0, 12}, {k, 0, 2}]]]] (*  array *)
Flatten[Table[t[n, k], {n, 0, 20}, {k, 0, 2}]] (*  A247309 *)

Let F = A000045 (Fibonacci numbers); then
(row 0, the bottom row) = (F(2n)), n >= 0;
(row 1, the middle row) = (F(2n)), n >= 0;
(row 2, the top row) = (F(2n-1)), n >= 0.
(n-th column sum) = (F(2n+2)), n >= 0.

A247310 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), (1,2), (1,-1), (1,-2), where no segment is followed by a segment in the same direction.

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 2, 1, 0, 0, 2, 3, 5, 3, 0, 0, 5, 8, 13, 8, 0, 0, 13, 21, 34, 21, 0, 0, 34, 55, 89, 55, 0, 0, 89, 144, 233, 144, 0, 0, 233, 377, 610, 377, 0, 0, 610, 987, 1597, 987, 0, 0, 1597, 2584, 4181, 2584, 0, 0, 4181, 6765, 10946, 6765, 0, 0, 10946

Offset: 0

Clark Kimberling, Sep 12 2014

Also, T(n,k) = number of strings s(0)..s(n) of integers such that s(0) = 0, s(n) = k, s(i) is in {0,1,2} for i = 0..n, and for i > 0, s(i) - s(i-1) is in {1,2} if i is odd, and s(i) - s(i-1) is in {-1,-2} if i is even. Every row of T consists of Fibonacci numbers, and (sum of numbers in column n) = A000045(n+1).

			First 10 columns:
0 .. 1 .. 0 .. 3 .. 0 .. 8 .. 0 ... 21 .. 0 ... 55
0 .. 1 .. 1 .. 2 .. 3 .. 5 .. 8 ... 13 .. 21 .. 34
1 .. 0 .. 2 .. 0 .. 5 .. 0 .. 13 .. 0 ... 34 .. 0
T(3,2) counts these 3 paths, given as vector sums applied to (0,0):
(1,2) + (1,-1) + (1,1);
(1,1) + (1,-1) + (1,2);
(1,2) + (1,-2) + (1,2).

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

Cf. A247049, A247309, A000045.

t[0, 0] = 1; t[0, 1] = 0; t[0, 2] = 0;
t[1, 0] = 0; t[1, 1] = 1; t[1, 2] = 1;
t[2, 0] = 2; t[2, 1] = 1; t[2, 2] = 0;
t[n_, 0] := If[OddQ[n], 0 , t[n - 1, 1] + t[n - 1, 2]]
t[n_, 1] := If[OddQ[n], t[n - 1, 0] , t[n - 1, 2]]
t[n_, 2] := If[OddQ[n], t[n - 1, 0] + t[n - 1, 1], 0]
TableForm[Reverse[Transpose[Table[t[n, k], {n, 0, 12}, {k, 0, 2}]]]] (*  array *)
u = Flatten[Table[t[n, k], {n, 0, 25}, {k, 0, 2}]] (*  A247310 *)

Let F = A000045 (Fibonacci numbers); then
(row 0, the bottom row):  F(1), 0 , F(3), 0 , F(5), 0, ...
(row 1, the middle row):  F(0), F(1), F(2), F(3), F(4), F(5), ...
(row 2, the top row):  0, F(2), 0, F(4), 0, F(6) , 0, ...

A247352 Rectangular array read by upwards by columns: T = T(n,k) = number of paths from (0,1) to (n,k), where 0 <= k <= 3, consisting of segments given by the vectors (1,1), (1,2), (1,-1).

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 1, 1, 0, 2, 2, 1, 2, 2, 3, 4, 2, 5, 8, 5, 5, 10, 12, 13, 10, 17, 28, 22, 17, 38, 49, 45, 38, 66, 100, 87, 66, 138, 191, 166, 138, 257, 370, 329, 257, 508, 724, 627, 508, 981, 1392, 1232, 981, 1900, 2721, 2373, 1900, 3702, 5254, 4621, 3702

Offset: 0

Clark Kimberling, Sep 15 2014

Also, T(n,k) = number of strings s(0)..s(n) of integers such that s(0) = 1, 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 .. 1 .. 1 .. 4 .. 5 .. 13 .. 22 .. 45 .. 87 ... 166
0 .. 1 .. 2 .. 3 .. 8 .. 12 .. 28 .. 49 .. 100 .. 191
1 .. 0 .. 2 .. 2 .. 5 .. 10 .. 17 .. 38 .. 66 ... 138
0 .. 1 .. 0 .. 2 .. 2 .. 5 ... 10 .. 17 .. 38 ... 66
T(5,0) counts these 5 paths, given as vector sums applied to (0,1):
(1,1) + (1,1) + (1,-1) + (1,-1) + (1 -1)
(1,1) + (1,-1) + (1,1) + (1,-1) + (1,-1)
(1,-1) + (1,1) + (1,1) + (1,-1) + (1,-1)
(1,1) + (1,-1) + (1,-1) + (1,1) + (1,-1)
(1,-1) + (1,1) + (1,-1) + (1,1) + (1,-1)
Partial sums of second components in each vector sum give the 3 integer strings described in comments:
(1,2,3,2,1,0),
(1,2,1,2,1,0),
(1,0,1,2,1,0),
(1,2,1,0,1,0),
(1,0,1,0,1,0).

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

Cf. A247049, A247321, A247353, A247354, A247355.

z = 50; t[0, 0] = 0; t[0, 1] = 1; t[0, 2] = 0; t[0, 3] = 0;
t[1, 3] = 1; 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}]] (* A247352 *)
TableForm[Reverse[Transpose[Table[t[n, k], {n, 0, 12}, {k, 0, 3}]]]] (* array  *)
v = Map[Total, u1]  (* A247353 *)
Table[t[n, 0], {n, 0, z}]   (* row 0, A247354*)
Table[t[n, 1], {n, 0, z}]   (* row 1, cf. row 0 *)
Table[t[n, 2], {n, 0, z}]   (* row 2, A247355 *)
Table[t[n, 3], {n, 0, z}]   (* row 3, A247325 *)

The four rows and 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):  0,1,0,2; x - x^3
(row 1):  1,0,0,2; 1 - x^2
(row 2):  0,1,2,3; x +2*x^2
(row 3):  0,1,1,4; x + x^2 + x^3
(n-th column sum) = 1,3,5,11; 1 + 3*x + 2*x^2.

A247323 Number of paths from (0,0) to (n,0), with vertices (i,k) satisfying 0 <= k <= 3, consisting of segments given by the vectors (1,1), (1,2), (1,-1).

Original entry on oeis.org

1, 0, 1, 1, 2, 5, 7, 18, 29, 63, 116, 229, 445, 856, 1677, 3229, 6298, 12185, 23675, 45922, 89097, 172931, 335460, 651065, 1263145, 2451184, 4756105, 9228777, 17907538, 34747357, 67424063, 130828370, 253859365, 492585879, 955810772, 1854647997, 3598744709

Offset: 0

Clark Kimberling, Sep 13 2014

Also, a(n) = number of strings s(0)..s(n) of integers such that s(0) = 0, s(n) = 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) = row 0 (the bottom row) of the array at A247321, and a(n+1) = row 1 of the same array.

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

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

Cf. A247049, A247321, A247322, 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];
Table[t[n, 0], {n, 0, z}];  (* A247323 *)

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

A247051 Rectangular array read upwards by columns: T = T(n,k) = number of paths from (0,2) to (n,k), where 0 <= k <= 2, consisting of segments given by the vectors (1,1), (1,2), (1,-1).

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 1, 2, 1, 2, 1, 4, 3, 4, 4, 5, 4, 9, 8, 9, 12, 13, 12, 22, 21, 22, 33, 34, 33, 56, 55, 56, 88, 89, 88, 145, 144, 145, 232, 233, 232, 378, 377, 378, 609, 610, 609, 988, 987, 988, 1596, 1597, 1596, 2585, 2584, 2585, 4180, 4181

Offset: 0

Clark Kimberling, Sep 11 2014

Also, T(n,k) = number of strings s(0)..s(n) of integers such that s(0) = 2, s(n) = k, and if i > 0, then s(i) is in {0,1,2} and s(i) - s(i-1) is in {1,2,-1}. The column sums form the Fibonacci sequence (A000045).

This is a 3-rowed array read upwards by columns. - N. J. A. Sloane, Sep 14 2014

			First 10 columns:
1 .. 0 .. 1 .. 1 .. 2 .. 3 .. 5 .. 8 .. 13 .. 21
0 .. 1 .. 0 .. 2 .. 1 .. 4 .. 4 .. 9 .. 12 .. 22
0 .. 0 .. 1 .. 0 .. 2 .. 1 .. 4 .. 4 .. 9 ... 12
T(3,1) counts these 2 paths, given as vector sums applied to (0,2): (1,-1) + (1,1) + (1,-1) and (1,-1) + (1,-1) + (1,1).
Partial sums of second components in each vector sum give the 2 integer strings described in Comments:  (2,1,2,1), (2,1,0,1).

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

Cf. A000045, A247049, A247050.

t[0, 0] = 0; t[0, 1] = 0; t[0, 2] = 1; 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]; TableForm[ Reverse[Transpose[Table[t[n, k], {n, 0, 12}, {k, 0, 2}]]]]   (* array *)
Flatten[Table[t[n, k], {n, 0, 20}, {k, 0, 2}]]   (* sequence *)

Let F = A000045, the Fibonacci numbers. Then (row 0, the bottom row) = F(n-2) + (-1)^n for n >= 1; (row 1, the middle row) = F(n-1) - (-1)^n for n >=0; (row 2, the top row) = F(n+1) for n >= 1.

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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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.

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A247325 Number of paths from (0,0) to (n,2), with vertices (i,k) satisfying 0 <= k <= 3, consisting of segments given by the vectors (1,1), (1,2), (1,-1).

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A247326 Number of paths from (0,0) to (n,3), with vertices (i,k) satisfying 0 <= k <= 3, consisting of segments given by the vectors (1,1), (1,2), (1,-1).

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A247050 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), (1,2), (1,-1).

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A247309 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), (1,0), (1,-1), (1,-2).

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A247310 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), (1,2), (1,-1), (1,-2), where no segment is followed by a segment in the same direction.

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A247352 Rectangular array read by upwards by columns: T = T(n,k) = number of paths from (0,1) to (n,k), where 0 <= k <= 3, consisting of segments given by the vectors (1,1), (1,2), (1,-1).

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