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Number of walks on cubic lattice starting at (0,0,0), ending at (0,0,n), remaining in the first (nonnegative) octant and using steps which are permutations of (-2,1,2), (-1,0,2), (-1,1,1), (0,0,1).
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%I #16 May 13 2020 05:21:24

%S 1,1,3,9,45,243,1611,11565,92169,782775,7078995,67106313,663693597,

%T 6797495115,71796341151,778700005929,8645902267653,97997326844499,

%U 1131383216210067,13278579296016357,158171247347313933,1909500035430728655,23334309656230665411

%N Number of walks on cubic lattice starting at (0,0,0), ending at (0,0,n), remaining in the first (nonnegative) octant and using steps which are permutations of (-2,1,2), (-1,0,2), (-1,1,1), (0,0,1).

%H Alois P. Heinz, <a href="/A328267/b328267.txt">Table of n, a(n) for n = 0..851</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path">Lattice path</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Self-avoiding_walk">Self-avoiding walk</a>

%p b:= proc(l) option remember; `if`(l[-1]=0, 1, (r-> add(

%p add(add(`if`(i+j+k=1, (h-> `if`(h[1]<0, 0, b(h)))(

%p sort(l-[i, j, k])), 0), k=r), j=r), i=r))([$-2..2]))

%p end:

%p a:= n-> b([0$2, n]):

%p seq(a(n), n=0..25);

%t b[l_] := b[l] = If[Last[l] == 0, 1, Function[r, Sum[If[i + j + k == 1, Function[h, If[h[[1]] < 0, 0, b[h]]][Sort[l - {i, j, k}]], 0], {i, r}, {j, r}, {k, r}]][Range[-2, 2]]];

%t a[n_] := b[{0, 0, n}];

%t a /@ Range[0, 25] (* _Jean-François Alcover_, May 13 2020, after Maple *)

%Y Cf. A002426, A328268.

%K nonn,walk

%O 0,3

%A _Alois P. Heinz_, Oct 10 2019