A188784 T(n,k) = Number of n-step self-avoiding walks on a k X k X k X k 4-cube summed over all starting positions.

1, 16, 0, 81, 64, 0, 256, 432, 192, 0, 625, 1536, 1944, 576, 0, 1296, 4000, 7936, 8928, 1536, 0, 2401, 8640, 22200, 41984, 39408, 4224, 0, 4096, 16464, 50112, 125840, 217728, 174720, 9984, 0, 6561, 28672, 98392, 296064, 702904, 1141248, 748128

Offset

1,2

Comments

If W is an n-step self-avoiding walk starting at (0,0,0,0), let D_i = max {x_i: x in W} - min {x_i: x in W} for i=1..4. Then if k >= max(D_i), (k-D_1)...(k-D_4) translates of W contribute to T(n,k). Therefore for each n, T(n,k) is a polynomial of degree 4 in k for k >= n-1. - Robert Israel, Apr 30 2018

Links

R. H. Hardin, Table of n, a(n) for n = 1..44

Formula

Empirical: T(1,k) = k^4.

Empirical: T(2,k) = 8*k^4 - 8*k^3.

Empirical: T(3,k) = 56*k^4 - 112*k^3 + 48*k^2 for k>1.

Empirical: T(4,k) = 392*k^4 - 1128*k^3 + 912*k^2 - 192*k for k>2.

Empirical: T(5,k) = 2696*k^4 - 9968*k^3 + 11424*k^2 - 4416*k + 384 for k>3.

Empirical: T(6,k) = 18584*k^4 - 82552*k^3 + 119616*k^2 - 64320*k + 9984 for k>4.

Empirical: T(7,k) = 127160*k^4 - 654960*k^3 + 1132704*k^2 - 762240*k + 162048 for k>5.

Empirical: T(8,k) = 871256*k^4 - 5064008*k^3 + 10076736*k^2 - 8024256*k + 2111616 for k>6.

Example

Table starts
.1....16.....81.....256....625...1296..2401..4096.6561
.0....64....432....1536...4000...8640.16464.28672
.0...192...1944....7936..22200..50112.98392
.0...576...8928...41984.125840.296064
.0..1536..39408..217728.702904
.0..4224.174720.1141248
.0..9984.748128
.0.24576

Crossrefs

Sequence in context: A331140 A059681 A135925 * A123935 A169764 A002607

Adjacent sequences: A188781 A188782 A188783 * A188785 A188786 A188787

Keyword

nonn,tabl

Author

R. H. Hardin, Apr 10 2011

Status

approved