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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. 3
1, 6, 54, 384, 2550, 16506, 105840, 677088, 4335606, 27829230, 179161554, 1156987728, 7493841264, 48672149064, 316920674880, 2068273848384, 13525486999542, 88612412883030, 581503640659830, 3821691744347400, 25150239955660050, 165713382866931570 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

LINKS

Harry J. Smith and 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: A116138 A227268 A300583 * A043026 A125837 A065088

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

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Last modified October 20 18:19 EDT 2019. Contains 328269 sequences. (Running on oeis4.)