

A129703


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


0



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
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OFFSET

0,2


COMMENTS

For n > 0, a(n)1 is the sum of the nth 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, (33*n)*a(n)+(62*n)*a(1+n)+(3+n)*a(n+2)+6+4*n}.
G.f.: ((1/2)*i)*sqrt(t+1)/(t*sqrt(3*t1))(1/2)*(t+1)*(1+2*t)/((t1)*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, t1)), 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



