A247352 -id:A247352 - cp's OEIS frontend

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.

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).

A247354 Number of paths from (0,1) 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

0, 1, 0, 2, 2, 5, 10, 17, 38, 66, 138, 257, 508, 981, 1900, 3702, 7154, 13925, 26966, 52381, 101594, 197150, 382578, 742257, 1440440, 2794777, 5423256, 10522954, 20418882, 39620597, 76879298, 149176601, 289460206, 561667802, 1089854522, 2114747217

Offset: 0

Clark Kimberling, Sep 15 2014

Also, a(n) = number of strings s(0)..s(n) of integers such that s(0) = 1, 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 A247352, and a(n+1) = row 1 of the same array.

			a(5) counts these 5 paths, each represented by a vector sum applied to (0,1):
(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)

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

Cf. A247352, A247353, 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]
Table[t[n, 0], {n, 0, z}]   (* A247354*)

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

A247355 Number of paths from (0,1) 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, 2, 3, 8, 12, 28, 49, 100, 191, 370, 724, 1392, 2721, 5254, 10223, 19812, 38456, 74628, 144769, 280984, 545107, 1057862, 2052520, 3982816, 7728177, 14995626, 29097643, 56460416, 109556004, 212580908, 412491201, 800394316, 1553079415, 3013584442

Offset: 0

Clark Kimberling, Sep 15 2014

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

			a(3) counts these 3 paths, each represented by a vector sum applied to (0,1):
(1,1) + (1,1) + (1,-1);
(1,1) + (1,-1) + (1,1);
(1,-1) + (1,1) + (1,1).

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

Cf. A247352, A247353, A247354.

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];
Table[t[n, 2], {n, 0, z}]   (* A247355 *)

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

cp's OEIS Frontend

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A247354 Number of paths from (0,1) to (n,0), with vertices (i,k) satisfying 0 <= k <= 3, consisting of segments given by the vectors (1,1), (1,2), (1,-1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A247355 Number of paths from (0,1) to (n,2), with vertices (i,k) satisfying 0 <= k <= 3, consisting of segments given by the vectors (1,1), (1,2), (1,-1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula