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A064036 Number of walks of length n on cubic lattice, starting at origin, staying in first (nonnegative) octant. 3
1, 3, 12, 51, 234, 1110, 5460, 27405, 140490, 729918, 3845016, 20447658, 109801692, 593806356, 3234529584, 17715445605, 97567971930, 539701180590, 2998595422680, 16719506691030, 93559970043540, 525093580540620, 2955822168597480, 16680150247605390, 94365481922990460 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Robert Israel, Table of n, a(n) for n = 0..500

R. K. Guy, Catwalks, Sandsteps and Pascal Pyramids, J. Integer Seqs., Vol. 3 (2000), #00.1.6.

FORMULA

a(n) = sum_j[C(n, j)B(j)B(j+1)B(n-j)] where B(k)=C(k, [k/2])=A001405(k)

E.g.f.: (BesselI(0, 2*x)+BesselI(1, 2*x))^3. - Vladeta Jovovic, Apr 28 2003

From Vaclav Kotesovec, Jun 10 2019: (Start)

Recurrence: (n+1)*(n+2)*(n+3)*a(n) = 4*(5*n^2+10*n+3)*a(n-1) + 4*(n-1)*(10*n^2+10*n-9)*a(n-2) - 144*(n-2)*(n-1)*a(n-3) - 144*(n-3)*(n-2)*(n-1)*a(n-4).

a(n) ~ 6^(n + 3/2) / (Pi*n)^(3/2). (End)

EXAMPLE

a(2)=12 since a(1) is obviously 3 and from each of these three positions there are four possible steps which remain in the first octant.

MAPLE

S:= series((BesselI(0, 2*x)+BesselI(1, 2*x))^3, x, 101):

seq(simplify(coeff(S, x, n))*n!, n=0..100); # Robert Israel, Oct 10 2016

CROSSREFS

Cf. A064037. The two- and one-dimensional equivalents are A005566 and A001405. With no restriction on the walks, the number is 6^n, i.e. A000400.

Sequence in context: A151188 A151189 A199875 * A195255 A241468 A224914

Adjacent sequences:  A064033 A064034 A064035 * A064037 A064038 A064039

KEYWORD

nonn

AUTHOR

Henry Bottomley, Aug 23 2001

STATUS

approved

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Last modified August 8 19:29 EDT 2020. Contains 336298 sequences. (Running on oeis4.)