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A328269 Number of walks on cubic lattice starting at (0,0,0), ending at (0,n,n), 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 A328269 #26 Oct 15 2020 12:37:33
%S A328269 1,3,26,343,5594,103730,2094028,44889351,1006126370,23337166962,
%T A328269 556199376622,13550764116530,336190200180652,8468872074477060,
%U A328269 216120719672921820,5577150906683145103,145324963753397617230,3819107708757101038562,101122686499165125017886
%N A328269 Number of walks on cubic lattice starting at (0,0,0), ending at (0,n,n), 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 A328269 Alois P. Heinz, <a href="/A328269/b328269.txt">Table of n, a(n) for n = 0..639</a>
%H A328269 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path">Lattice path</a>
%H A328269 Wikipedia, <a href="https://en.wikipedia.org/wiki/Self-avoiding_walk">Self-avoiding walk</a>
%F A328269 a(n) = A328300(2n,n).
%F A328269 a(n) is odd <=> n in { A000225 }.
%F A328269 a(n) ~ c * 2^(3*n) * (2 + sqrt(3))^n / n^2, where c =
%F A328269 0.081957778985952080274457799679795068000445171394180053136120884510526907545... - _Vaclav Kotesovec_, May 10 2020
%e A328269 a(1) = 3: [(0,0,0),(1,0,0),(0,1,1)], [(0,0,0),(0,1,0),(0,1,1)], [(0,0,0),(0,0,1),(0,1,1)].
%e A328269 a(2) = 26: [(0,0,0),(1,0,0),(2,0,0),(1,1,1),(0,2,2)], [(0,0,0),(1,0,0),(1,1,0),(1,1,1),(0,2,2)], ..., [(0,0,0),(0,0,1),(0,1,1),(0,1,2),(0,2,2)], [(0,0,0),(0,0,1),(0,0,2),(0,1,2),(0,2,2)].
%p A328269 b:= proc(l) option remember; `if`(l[-1]=0, 1, (r-> add(
%p A328269       add(add(`if`(i+j+k=1, (h-> `if`(h[1]<0, 0, b(h)))(
%p A328269       sort(l-[i, j, k])), 0), k=r), j=r), i=r))([$-1..1]))
%p A328269     end:
%p A328269 a:= n-> b([0, n$2]):
%p A328269 seq(a(n), n=0..23);
%t A328269 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 A328269 a[n_] := b[{0, n, n}];
%t A328269 a /@ Range[0, 23] (* _Jean-François Alcover_, May 12 2020, after Maple *)
%Y A328269 Bisection (even part) of A328280.
%Y A328269 Cf. A000225, A000984, A277262, A328270, A328300.
%K A328269 nonn,walk
%O A328269 0,2
%A A328269 _Alois P. Heinz_, Oct 10 2019