A247322 -id:A247322 - 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.

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).

A247353 Number of paths from (0,1) 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, 3, 5, 11, 20, 40, 77, 149, 291, 561, 1094, 2116, 4113, 7975, 15477, 30035, 58268, 113084, 219397, 425753, 826091, 1602969, 3110382, 6035336, 11710993, 22723803, 44093269, 85558059, 166016420, 322136912, 625072109, 1212885517, 2353473731, 4566663857

Offset: 0

Clark Kimberling, Sep 15 2014

Also, a(n) = number of strings s(0)..s(n) of integers such that s(0) = 1, 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 A247352.

			a(2) counts these 5 paths, each represented by a vector sum applied to (0,1):
(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. A247322, A247352, 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 *)
u1 = Table[t[n, k], {n, 0, z}, {k, 0, 3}];
v = Map[Total, u1]  (* A247353 *)

A247353(n) = A247354(n) + A247354(n+1) + A247355(n) + A247321(n).

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

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).

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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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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A247353 Number of paths from (0,1) 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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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).

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