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

0,2

Comments

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

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.

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

Example

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

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 *)

Prog

(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

Crossrefs

Column k=3 of A225094.

Sequence in context: A227268 A364008 A300583 * A043026 A371397 A125837

Adjacent sequences: A060771 A060772 A060773 * A060775 A060776 A060777

Keyword

nonn,easy

Author

Len Smiley, Apr 25 2001

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

Status

approved