[seqfan] Re: Self-avoiding walks on nXnXn cubic lattice, guess the coefficients

[Ron Hardin]

The same thing happens in 4 dimensions

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

Table starts
 1 16 81 256 625 1296 2401 4096 6561 10000 14641 20736 28561 38416 50625
 0 64 432 1536 4000 8640 16464 28672 46656 72000 106480 152064 210912 285376 
378000
 0 192 1944 7936 22200 50112 98392 175104 289656 452800 676632 974592 1361464 
1853376 2467800
 0 576 8928 41984 125840 296064 597632 1084928 1821744 2881280 4346144 6308352 
8869328 12139904 16240320
 0 1536 39408 217728 702904 1726080 3583320 6635392 11307768 18090624 27538840 
40272000
 0 4224 174720 1141248 3974784 10183872 21725776 41004288 70869216 114616384 
175987632 259170816
 0 9984 748128 5886720 22274448 59693952 130988744 252065024 441881832 722450048 
1118832392 1659143424
 0 24576 3203232 30437760 125478336 352052352 794632800 1558721152 2770757712 
4578094656 7148994304 10672629120

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

The coefficients of k^4 are http://oeis.org/A010575
Number of n-step self-avoiding walks on 4-d cubic lattice


 rhhardin at mindspring.com
rhhardin at att.net (either)