A129703 Number of different walks generated by n steps that can only go in {east, southeast, southwest} directions on the 300-degree wedge in a 60-degree equilateral triangular lattice.

1, 3, 6, 14, 36, 97, 268, 751, 2124, 6047, 17304, 49722, 143366, 414585, 1201918, 3492118, 10165780, 29643871, 86574832, 253188112, 741365050, 2173243129, 6377181826, 18730782253, 55062586342, 161995031227, 476941691178

Offset

0,2

Comments

For n > 0, a(n)-1 is the sum of the n-th row of Motzkin's triangle (A026300). - Daniel Suteu, Feb 23 2018

Links

Table of n, a(n) for n=0..26.

Formula

Recurrence: {a(3) = 14, a(4) = 36, a(1) = 3, a(2) = 6, a(0) = 1, (-3-3*n)*a(n)+(-6-2*n)*a(1+n)+(3+n)*a(n+2)+6+4*n}.

G.f.: ((1/2)*i)*sqrt(t+1)/(t*sqrt(3*t-1))-(1/2)*(t+1)*(-1+2*t)/((t-1)*t).

a(n) = 1 + Sum_{k=0..n} Sum_{t=0..floor(k/2)} binomial(n, 2*t + n - k) * (binomial(2*t + n - k, t) - binomial(2*t + n - k, t-1)), for n > 0. - Daniel Suteu, Feb 23 2018

Example

a(1) = 3 because all three directions are permissible from the origin;
a(2) = 6 because all three directions are permissible following the southwestern step and the southwest as well as southeast steps are permissible following the southeastern step, but only the eastern step is permissible following one step east.

Crossrefs

Sequence in context: A332362 A196479 A147772 * A001550 A197461 A100446

Adjacent sequences: A129700 A129701 A129702 * A129704 A129705 A129706

Keyword

nonn,walk

Author

Rebecca Xiaoxi Nie (rebecca.nie(AT)utoronto.ca), Jun 01 2007

Status

approved