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A328280 Number of n-step walks on cubic lattice starting at (0,0,0), ending at (0,floor(n/2),ceiling(n/2)), 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 A328280 #19 May 12 2020 11:50:30
%S A328280 1,1,3,7,26,82,343,1257,5594,22411,103730,440350,2094028,9255877,
%T A328280 44889351,204385719,1006126370,4685719954,23337166962,110633755459,
%U A328280 556199376622,2674751727209,13550764116530,65935784179142,336190200180652,1651985253047884
%N A328280 Number of n-step walks on cubic lattice starting at (0,0,0), ending at (0,floor(n/2),ceiling(n/2)), 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 A328280 Alois P. Heinz, <a href="/A328280/b328280.txt">Table of n, a(n) for n = 0..1000</a>
%H A328280 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path">Lattice path</a>
%H A328280 Wikipedia, <a href="https://en.wikipedia.org/wiki/Self-avoiding_walk">Self-avoiding walk</a>
%F A328280 a(n) = A328300(n,floor(n/2)).
%p A328280 b:= proc(l) option remember; `if`(l[-1]=0, 1, (r-> add(
%p A328280       add(add(`if`(i+j+k=1, (h-> `if`(h[1]<0, 0, b(h)))(
%p A328280       sort(l-[i, j, k])), 0), k=r), j=r), i=r))([$-1..1]))
%p A328280     end:
%p A328280 a:= n-> (t-> b([0, t, n-t]))(iquo(n, 2)):
%p A328280 seq(a(n), n=0..31);
%t A328280 b[l_] := b[l] = If[Last[l] == 0, 1, Sum[If[i + j + k == 1, Function[h, If[h[[1]] < 0, 0, b[h]]][Sort[l - {i, j, k}]], 0], {i, {-1, 0, 1}}, {j, {-1, 0, 1}}, {k, {-1, 0, 1}}]];
%t A328280 a[n_] := With[{t = Quotient[n, 2]}, b[{0, t, n - t}]];
%t A328280 a /@ Range[0, 31] (* _Jean-François Alcover_, May 12 2020, after Maple *)
%Y A328280 Bisection gives A328269 (even part).
%Y A328280 Cf. A328281, A328300.
%K A328280 nonn,walk
%O A328280 0,3
%A A328280 _Alois P. Heinz_, Oct 10 2019