%I #19 Nov 25 2018 13:08:33
%S 1,3,6,14,36,97,268,751,2124,6047,17304,49722,143366,414585,1201918,
%T 3492118,10165780,29643871,86574832,253188112,741365050,2173243129,
%U 6377181826,18730782253,55062586342,161995031227,476941691178
%N 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.
%C For n > 0, a(n)-1 is the sum of the n-th row of Motzkin's triangle (A026300). - _Daniel Suteu_, Feb 23 2018
%F 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}.
%F G.f.: ((1/2)*i)*sqrt(t+1)/(t*sqrt(3*t-1))-(1/2)*(t+1)*(-1+2*t)/((t-1)*t).
%F 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
%e a(1) = 3 because all three directions are permissible from the origin;
%e 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.
%K nonn,walk
%O 0,2
%A Rebecca Xiaoxi Nie (rebecca.nie(AT)utoronto.ca), Jun 01 2007