A247629 -id:A247629 - cp's OEIS frontend

A247630 Number of paths from (0,0) to the line x = n, each segment given by a vector (1,1), (1,-1), or (2,0), not crossing the x-axis, and including no horizontal segment on the x-axis.

1, 1, 2, 4, 10, 20, 50, 104, 258, 552, 1362, 2972, 7306, 16172, 39650, 88720, 217090, 489872, 1196834, 2719028, 6634890, 15157188, 36949266, 84799992, 206549250, 475894200, 1158337650, 2677788492, 6513914634, 15102309468, 36718533570, 85347160608

Offset: 0

Clark Kimberling, Sep 21 2014

a(n) = sum of the numbers in row n of the triangle at A247629.

			First 9 rows:
1
0 ... 1
1 ... 0 ... 1
0 ... 3 ... 0 ... 1
4 ... 0 ... 5 ... 0 ... 1
0 ... 12 .. 0 ... 7 ... 0 ...1
16 .. 0 ... 24 .. 0 ... 9 ... 0 ... 1
0 ... 52 .. 0 ... 40 .. 0 ... 11 .. 0 ... 1
68 .. 0 ... 116 . 0 ... 60 .. 0 ... 13 .. 0 ... 1
T(4,2) counts these 5 paths given as vector sums applied to (0,0):
(1,1) + (1,1) + (1,1) + (1,-1)
(1,1) + (1,1) + (2,0)
(1,1) + (1,1) + (1,-1) + (1,1)
(1,1) + (2,0) + (1,1)
(1,1) + (1,-1) + (1,1) + (1,-1)
a(4) = 4 + 0 + 5 + 0 + 1 = 10.

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

Cf. A247629, A247623.

t[0, 0] = 1; t[1, 1] = 1; t[2, 0] = 1; t[2, 2] = 1; t[n_, k_] := t[n, k] = If[n >= 2 && k >= 1,    t[n - 1, k - 1] + t[n - 1, k + 1] + t[n - 2, k], 0]; t[n_, 0] := t[n, 0] = If[n >= 2, t[n - 2, 0] + t[n - 1, 1], 0]; u = Table[t[n, k], {n, 0, 16}, {k, 0, n}]; TableForm[u] (* A247629 array *)
v = Flatten[u] (* A247629 sequence *)
Map[Total, u] (* A247630 *)

Conjecture: -(n+1)*(n-2)*a(n) -(n-1)*(n-4)*a(n-1) +2*(3*n-2)*(n-2)*a(n-2) +2*(3*n-5)*(n-3)*a(n-3) +(-n^2+7*n-2)*a(n-4) -(n-1)*(n-6)*a(n-5)=0. - R. J. Mathar, Sep 23 2014

A247622 Triangular array: T(n,k) = number of paths from (0,0) to (n,k), each segment given by a vector (1,1), (1,-1), or (2,0), not crossing the x-axis, and including no horizontal segment on the x-axis.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 3, 0, 1, 3, 0, 5, 0, 1, 0, 11, 0, 7, 0, 1, 11, 0, 23, 0, 9, 0, 1, 0, 45, 0, 39, 0, 11, 0, 1, 45, 0, 107, 0, 59, 0, 13, 0, 1, 0, 197, 0, 205, 0, 83, 0, 15, 0, 1, 197, 0, 509, 0, 347, 0, 111, 0, 17, 0, 1, 0, 903, 0, 1061, 0, 541, 0, 143, 0

Offset: 0

Clark Kimberling, Sep 21 2014

			First 9 rows:
1
0 ... 1
1 ... 0 ... 1
0 ... 3 ... 0 ... 1
3 ... 0 ... 5 ... 0 ... 1
0 ... 11 .. 0 ... 7 ... 0 ...1
11 .. 0 ... 23 .. 0 ... 9 ... 0 ... 1
0 ... 45 .. 0 ... 39 .. 0 ... 11 .. 0 ... 1
45 .. 0 ... 107 . 0 ... 59 .. 0 ... 13 .. 0 ... 1
T(3,1) counts these 3 paths given as vector sums applied to (0,0):
(1,1) + (1,-1), (2,0), (1,-1) + (1,1).

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

Cf. A247623, A247629, A026300, A001003 (1st column of this triangle).

t[0, 0] = 1; t[1, 1] = 1; t[2, 0] = 1; t[2, 2] = 1; t[n_, k_] := t[n, k] = If[n >= 2 && k >= 1, t[n - 1, k - 1] + t[n - 1, k + 1] + t[n - 2, k], 0]; t[n_, 0] := t[n, 0] = t[n - 1, 1]; u = Table[t[n, k], {n, 0, 16}, {k, 0, n}];
v = Flatten[u] (* A247622 sequence *)
TableForm[u]   (* A247622 array *)
Map[Total, u]  (* A247623 *)

cp's OEIS Frontend

A247630 Number of paths from (0,0) to the line x = n, each segment given by a vector (1,1), (1,-1), or (2,0), not crossing the x-axis, and including no horizontal segment on the x-axis.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula