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A328295 Number of n-step walks on cubic lattice starting at (0,0,0), remaining in the first (nonnegative) octant and using steps (0,0,1), (0,1,0), (1,0,0), (-1,1,1), (1,-1,1), and (1,1,-1).

%I A328295 #13 May 13 2020 05:20:35
%S A328295 1,3,12,57,294,1590,8856,50301,289590,1683306,9853320,57977922,
%T A328295 342494118,2029350972,12052664868,71715479535,427347761010,
%U A328295 2549540104944,15224944518084,90988367614254,544115710748898,3255541325220204,19486893225315138,116685749052336714
%N A328295 Number of n-step walks on cubic lattice starting at (0,0,0), remaining in the first (nonnegative) octant and using steps (0,0,1), (0,1,0), (1,0,0), (-1,1,1), (1,-1,1), and (1,1,-1).
%H A328295 Alois P. Heinz, <a href="/A328295/b328295.txt">Table of n, a(n) for n = 0..630</a>
%H A328295 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path">Lattice path</a>
%H A328295 Wikipedia, <a href="https://en.wikipedia.org/wiki/Self-avoiding_walk">Self-avoiding walk</a>
%p A328295 b:= proc(l) option remember; `if`(l[-1]=0, 1, (r-> add(
%p A328295       add(add(`if`(i+j+k=1, (h-> `if`(h[1]<0, 0, b(h)))(
%p A328295       sort(l-[i, j, k])), 0), k=r), j=r), i=r))([$-1..1]))
%p A328295     end:
%p A328295 a:= n-> add(add(b(sort([i, j, n-i-j])), j=0..n-i), i=0..n):
%p A328295 seq(a(n), n=0..23);
%t A328295 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}]][{-1, 0, 1}]];
%t A328295 a[n_] := Sum[Sum[b[Sort[{i, j, n - i - j}]], {j, 0, n - i}], {i, 0, n}];
%t A328295 a /@ Range[0, 23] (* _Jean-François Alcover_, May 13 2020, after Maple *)
%Y A328295 Row sums of A328297.
%K A328295 nonn,walk
%O A328295 0,2
%A A328295 _Alois P. Heinz_, Oct 11 2019