A060774 a(n) = number of lattice paths from (0,0,0) to (n,n,n) along the cracks on the surface of a Rubik-ized n X n X n cube so that no step increases distance from goal.

1, 6, 54, 384, 2550, 16506, 105840, 677088, 4335606, 27829230, 179161554, 1156987728, 7493841264, 48672149064, 316920674880, 2068273848384, 13525486999542, 88612412883030, 581503640659830, 3821691744347400, 25150239955660050, 165713382866931570

Offset: 0

Author

Len Smiley, Apr 25 2001

Keywords

• nonn
• easy

Comments

3-dimensional version of block-walking (0,0) to (n,n) in binomial(2n,n) ways.

Examples

			a(1)=6: XYZ, XZY, YXZ, YZX, ZXY, ZYX.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000 (terms n = 1..200 from Harry J. Smith)
• W. Li and E. T. H. Wang, A bug's shortest path on a cube, Mathematics Magazine 58:4 (Sept. 1985), pp. 219-221.

Crossrefs

Column k=3 of A225094.

Programs

Maple

A060774 := proc(n)
        `if`(n=0, 1,
        6*(binomial(3*n,n)-binomial(2*n,n)) ) ;
end proc: # R. J. Mathar, Oct 31 2015

Mathematica

Rest[CoefficientList[Series[-(6/Sqrt[1-4z])+(12Cos[ArcCos[1-27z/2]/6])/Sqrt[4-27z], {z, 0, 20}], z]] (* Benedict W. J. Irwin, Jul 12 2016 *)

PARI

j=[]; for(n=1,50,j=concat(j,6*(binomial(3*n,n)-binomial(2*n,n)))); j

PARI

{ for (n=1, 200, write("b060774.txt", n, " ", 6*(binomial(3*n, n) - binomial(2*n, n))); ) } \\ Harry J. Smith, Jul 11 2009

Formula

a(n) = 6*binomial(3n, n) - 6*binomial(2n, n).

a(n) = 6*A000846(n) for n>0. - R. J. Mathar, Oct 31 2015

Conjecture: 2*n*(2*n-1)*(n-1)*a(n) + (n-1)*(13*n^2-209*n+258)*a(n-1) + 2*(-259*n^3+1785*n^2-3728*n+2460)*a(n-2) + 6*(295*n^3-2068*n^2+4833*n-3780)*a(n-3) - 36*(3*n-10)*(2*n-7)*(3*n-11)*a(n-4) = 0. - R. J. Mathar, Oct 31 2015

Conjecture: 2*n*(n-1)*(2*n-1)*(11*n^2-33*n+24)*a(n) - (n-1)*(473*n^4-1892*n^3+2561*n^2-1338*n+216)*a(n-1) + 6*(3*n-5)*(3*n-4)*(2*n-3)*(11*n^2-11*n+2)*a(n-2) = 0. - R. J. Mathar, Oct 31 2015

From Benedict W. J. Irwin, Jul 12 2016: (Start)

G.f.: -6/sqrt(1-4*x) + 12*cos(arccos(1-27*x/2)/6)/sqrt(4-27*x).

E.g.f: -6*E^(2*x)*BesselI(0,2*x) + 6*2F2(1/3,2/3;1/2,1;27*x/4).

(End)

a(n) ~ 4^(-n)*(3^(3*n+3/2))/sqrt(Pi*n). - Ilya Gutkovskiy, Jul 12 2016

Extensions

Corrected by Franklin T. Adams-Watters and T. D. Noe, Oct 25 2006

a(0)=1 prepended by Alois P. Heinz, Sep 09 2016