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).

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

0,5

Comments

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.

Links

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

Formula

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).

Example

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).

Mathematica

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 *)

Crossrefs

Cf. A247049, A247321, A247322, A247325, A247326.

Sequence in context: A303802 A045357 A168035 * A099357 A306918 A027038

Adjacent sequences: A247320 A247321 A247322 * A247324 A247325 A247326

Keyword

nonn,easy

Author

Clark Kimberling, Sep 13 2014

Status

approved