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
KEYWORD
nonn,tabl
AUTHOR
R. H. Hardin, Apr 10 2011
STATUS
approved