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%I #15 May 13 2020 07:02:37
%S 1,2,9,28,130,492,2401,10056,50346,224110,1141030,5284200,27222364,
%T 129582278,673340265,3270171504,17104148290,84342959172,443406172278,
%U 2212675109180,11680186909062,58844537998598,311667574680190,1582458820299408,8404755004516300
%N Total number of nodes in all 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 Alois P. Heinz, <a href="/A328281/b328281.txt">Table of n, a(n) for n = 0..1000</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>
%F a(n) = (n+1) * A328280(n).
%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))([$-1..1]))
%p end:
%p a:= n-> (t-> (n+1)*b([0, t, n-t]))(iquo(n, 2)):
%p seq(a(n), n=0..31);
%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}]][{-1, 0, 1}]];
%t a[n_] := Function[t, (n + 1) b[{0, t, n - t}]][Quotient[n, 2]];
%t a /@ Range[0, 31] (* _Jean-François Alcover_, May 13 2020, after Maple *)
%Y Cf. A328280.
%K nonn,walk
%O 0,2
%A _Alois P. Heinz_, Oct 10 2019
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