%I A129637
%S A129637 1,3,11,41,157,607,2367,9277,36505,144059,569779,2257521,8957109,
%T A129637 35579351,141460391,562871557,2241129905,8928207987,35584894299,
%U A129637 141886838329,565938926669
%N A129637 Number of n-step paths that can go {west, south-east, south-west,north-west} 
               on a 240 degree wedge on the equilateral triangular lattice.
%C A129637 If we use the "hour hands" 1,3,5,7,9,11 on the 12 hour clock to specify 
               directions in the triangular lattice, the allowable steps are in 
               directions 5,7,9,11 and the path is restricted to stay on or above 
               the 1-7 line. In the Mathematica recurrence below, a(n,k) denotes 
               the number of paths of length n ending k units from the 1-7 line, 
               counted by the last step. - David Callan (callan(AT)stat.wisc.edu), 
               Jul 22 2008
%F A129637 Recurrence: (-28-28*n)*a(n)+(-13-n)*a(1+n)+(21+6*n)*a(n+2)+(-4-n)*a(n+3), 
               a(0) = 1, a(1) = 3, a(2) = 11
%F A129637 Generating function = (1/4*i)*sqrt(-1+2*t+7*t^2)/((-1+4*t)*t)-(1/4)*(-1+5*t)/
               (t*(-1+4*t))
%F A129637 Differential equation: -(1+28*t^3-6*t+t^2)*t*(diff(f(t), t))+(9*t-12*t^2-1-28*t^3)*f(t)+1-3*t, 
               f(0) = 1
%e A129637 a(1) = 3 because only 3 out of the 4 steps are permissible from the origin;
%e A129637 a(2) = 11 because the north-west and west steps are followed by 4 permissible 
               steps each, but the south-west step is only followed by 3 permissible 
               steps.
%t A129637 a[0,0]=1; a[n_,k_]/;k<0 || k>n := 0; a[n_,k_]/;0<=k<=n := a[n,k] = 2a[n-1,
               k-1] + a[n-1,k] + a[n-1,k+1]; a[n_]:=Sum[a[n,k],{k,0,n}]; Table[a[n],
               {n,0,10}] - David Callan (callan(AT)stat.wisc.edu), Jul 22 2008
%Y A129637 Cf. A129400.
%Y A129637 Sequence in context: A079935 A113437 A076540 this_sequence A084077 A027103 
               A151086
%Y A129637 Adjacent sequences: A129634 A129635 A129636 this_sequence A129638 A129639 
               A129640
%K A129637 nonn
%O A129637 0,2
%A A129637 Rebecca Xiaoxi Nie (rebecca.nie(AT)utoronto.ca), May 31 2007