A151087 - OEIS

%I #16 Apr 21 2024 22:20:25

%S 1,3,11,41,165,687,2951,12861,56937,255227,1156995,5287185,24325901,

%T 112587815,523956543,2449799269,11500087377,54174459251,256027253243,

%U 1213527789561,5767023994869,27470856080927,131136432604919,627242243728269,3005666150390713,14426914113698667,69354696385966451

%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, -1), (-1, 1, 1), (0, 0, 1), (1, 0, 0), (1, 0, 1)}.

%H Robert Israel, <a href="/A151087/b151087.txt">Table of n, a(n) for n = 0..234</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.

%F From _Vaclav Kotesovec_, May 01 2018: (Start)

%F Recurrence: (n+1)*(n+2)^2*(2048*n^4 - 16640*n^3 + 43280*n^2 - 34992*n - 2979)*a(n) = (n+1)*(20480*n^6 - 123392*n^5 + 103072*n^4 + 436304*n^3 - 547134*n^2 - 114375*n - 35748)*a(n-1) - (71680*n^7 - 578304*n^6 + 1432624*n^5 - 601872*n^4 - 2284697*n^3 + 3374358*n^2 - 1558113*n + 83412)*a(n-2) + 2*(61440*n^7 - 804352*n^6 + 4058592*n^5 - 9580432*n^4 + 8510358*n^3 + 5565239*n^2 - 15547821*n + 7808040)*a(n-3) + (n-3)*(411648*n^6 - 3619072*n^5 + 8794512*n^4 + 1785776*n^3 - 28370523*n^2 + 22689914*n + 2453460)*a(n-4) - 5*(n-4)*(n-3)*(618496*n^5 - 4658688*n^4 + 10072288*n^3 - 1656912*n^2 - 13667306*n + 8981007)*a(n-5) + 1275*(n-5)*(n-4)*(n-3)*(2048*n^4 - 8448*n^3 + 5648*n^2 + 9840*n - 9283)*a(n-6).

%F a(n) ~ 5^(n+1) / (Pi*sqrt(2)*n) * (1 + sqrt(5*Pi/(2*n))/4).

%F (End)

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

%p f:= proc(n,p) option remember; local s, r;

%p   if n <= min(p) then return 5^n fi;

%p   if min(p)>= 1 then add(procname(n-1,p+s),s=Steps)

%p   elif p[1]>= 1 then add(procname(n-1,p+s),s=Steps[2..5])

%p   else add(procname(n-1,p+s), s=Steps[3..5])

%p   fi

%p end proc:

%p map(f, [$0..30]); # _Robert Israel_, May 01 2018

%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, j, -1 + k, -1 + n] + aux[-1 + i, j, k, -1 + n] + aux[i, j, -1 + k, -1 + n] + aux[1 + i, -1 + j, -1 + k, -1 + n] + aux[1 + i, 1 + j, 1 + 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 0,2

%A _Manuel Kauers_, Nov 18 2008