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 A151327 Number of walks within N^2 (the first quadrant of Z^2) starting at (0,0) and consisting of n steps taken from {(-1, 1), (-1, 0), (0, 1), (1, -1), (1, 0), (1, 1)} 1

%I

%S 1,3,15,76,413,2281,12889,73541,423921,2458383,14335834,83922633,

%T 492956132,2903156720,17135951352,101330250964,600140389918,

%U 3559105598556,21131319068601,125585737386758,747013179830622,4446753991483192,26487831271866795,157871848076357815,941434100552046728

%N Number of walks within N^2 (the first quadrant of Z^2) starting at (0,0) and consisting of n steps taken from {(-1, 1), (-1, 0), (0, 1), (1, -1), (1, 0), (1, 1)}

%H Robert Israel, <a href="/A151327/b151327.txt">Table of n, a(n) for n = 0..400</a>

%H A. Bostan and M. Kauers, <a href="https://arxiv.org/abs/0811.2899">Automatic Classification of Restricted Lattice Walks</a>, arXiv:0811.2899 [math.CO], 2008-2009.

%H M. Bousquet-MÃ©lou and M. Mishna, <a href="https://arxiv.org/abs/0810.4387">Walks with small steps in the quarter plane</a>, arXiv:0810.4387 [math.CO], 2008-2009.

%p F:= proc(x, y, n) option remember; local t, s, u;

%p t:= 0:

%p if n <= min(x, y) then return 6^n fi;

%p for s in [[-1, 1], [-1, 0], [0, 1], [1, -1], [1, 0], [1, 1]] do

%p u:= [x, y]+s;

%p if min(u) >= 0 then t:= t + procname(op(u), n-1) fi

%p od;

%p t

%p end proc:

%p seq(F(0, 0, n), n=0..40); # _Robert Israel_, Jun 05 2018

%t aux[i_Integer, j_Integer, n_Integer] := Which[Min[i, j, n] < 0 || Max[i, j] > n, 0, n == 0, KroneckerDelta[i, j, n], True, aux[i, j, n] = aux[-1 + i, -1 + j, -1 + n] + aux[-1 + i, j, -1 + n] + aux[-1 + i, 1 + j, -1 + n] + aux[i, -1 + j, -1 + n] + aux[1 + i, -1 + j, -1 + n] + aux[1 + i, j, -1 + n]]; Table[Sum[aux[i, j, n], {i, 0, n}, {j, 0, n}], {n, 0, 25}]

%K nonn,walk

%O 0,2

%A _Manuel Kauers_, Nov 18 2008

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Last modified April 18 02:38 EDT 2021. Contains 343072 sequences. (Running on oeis4.)