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).

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

Author

Clark Kimberling, Sep 15 2014

Keywords

• nonn
• easy

Comments

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.

Examples

			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)
		

Links

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

Crossrefs

Cf. A247352, A247353, A247355.

Programs

Mathematica

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*)

Formula

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).