A149595 - OEIS

%I #12 Nov 04 2014 17:34:22

%S 1,1,5,15,59,219,921,3729,15511,66307,285723,1231425,5377117,23746851,

%T 104998531,465502605,2084418377,9360389879,42066742411,189986019637,

%U 861969546027,3914101686361,17807890803795,81343463279557,372217083213775,1704575842549479,7825997996053383,36010679557986875

%N Number of walks within N^3 (the first octant of Z^3) starting at (0,0,0) and consisting of n steps taken from {(-1, -1, 0), (-1, 0, -1), (0, -1, 0), (1, 1, -1), (1, 1, 1)}.

%H Alois P. Heinz, <a href="/A149595/b149595.txt">Table of n, a(n) for n = 0..100</a>

%H A. Bostan and M. Kauers, <a href="http://arxiv.org/abs/0811.2899">Automatic Classification of Restricted Lattice Walks</a>, arXiv:0811.2899 [math.CO], 2008.

%p N:= 30: # to get a(0) to a(N)

%p steps:= [[-1,-1,0],[-1,0,-1],[0,-1,0],[1,1,-1],[1,1,1]]:

%p P[0]:= {[0,0,0]}:

%p A[0]:= 1:

%p B[0,[0,0,0]]:= 1:

%p for n from 1 to N do

%p   A[n]:= 0:

%p   P[n]:= {}:

%p   for p in P[n-1] do

%p     for s in steps do

%p       pp:= p + s;

%p       if min(pp) < 0 then next fi;

%p       P[n]:= P[n] union {pp};

%p       A[n]:= A[n] + B[n-1,p];

%p       if assigned(B[n,pp]) then B[n,pp]:= B[n,pp] + B[n-1,p]

%p       else B[n,pp]:= B[n-1,p]

%p       fi;

%p     od

%p   od

%p od:

%p seq(A[n],n=0..N); # _Robert Israel_, Nov 03 2014

%p # second Maple program:

%p b:= proc(n, l) option remember; `if`(n=0, 1, add((p->

%p       `if`(min(p[])<0, 0, b(n-1, p)))(l+h), h=[[-1$2, 0],

%p        [-1, 0, -1], [0, -1, 0], [1$2, -1], [1$3]]))

%p     end:

%p a:= n-> b(n, [0$3]):

%p seq(a(n), n=0..30);  # _Alois P. Heinz_, Nov 04 2014

%t aux[i_Integer, j_Integer, k_Integer, n_Integer] := Which[Min[i, j, k, n] < 0 || Max[i, j, k] > n, 0, n == 0, KroneckerDelta[i, j, k, n], True, aux[i, j, k, n] = aux[-1 + i, -1 + j, -1 + k, -1 + n] + aux[-1 + i, -1 + j, 1 + k, -1 + n] + aux[i, 1 + j, k, -1 + n] + aux[1 + i, j, 1 + k, -1 + n] + aux[1 + i, 1 + j, k, -1 + n]]; Table[Sum[aux[i, j, k, n], {i, 0, n}, {j, 0, n}, {k, 0, n}], {n, 0, 10}]

%K nonn,walk

%O 0,3

%A _Manuel Kauers_, Nov 18 2008