A328268 - OEIS

%I #15 May 13 2020 07:02:31

%S 1,2,9,36,225,1458,11277,92520,829521,7827750,77868945,805275756,

%T 8628016761,95164931610,1076945117265,12459200094864,146980338550101,

%U 1763951883200982,21496281107991273,265571585920327140,3321596194293592593,42009000779476030410

%N Total number of nodes in all 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="/A328268/b328268.txt">Table of n, a(n) for n = 0..848</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) * A328267(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))([$-2..2]))

%p     end:

%p a:= n-> (n+1)*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_] := (n + 1) b[{0, 0, n}];

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

%Y Cf. A328267.

%K nonn,walk

%O 0,2

%A _Alois P. Heinz_, Oct 10 2019