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

[Ron Hardin]

Updated with fuller explicit table, two more polynomial rows

Number of k-step self-avoiding walks on a nXnXn cubic lattice summed over all 
starting positions

T(k,n) Table starts
Table starts
.1...8.....27......64......125......216.......343.......512.......729......1000
.0..24....108.....288......600.....1080......1764......2688......3888......5400
.0..48....342....1056.....2370.....4464......7518.....11712.....17226.....24240
.0..96...1104....3984.....9612....18888.....32712.....51984.....77604....110472
.0.144...3240...14256....37470....77184....137754....223536....338886....488160
.0.240...9504...51504...148224...320328....588924....975216...1500408...2185704
.0.192..25344..177120...568248..1298016...2466510...4175136...6525450...9619008
.0.144..67824..608928..2188608..5299056..10416624..18026640..28617228..42676728
.0...0.167016.2013360..8227752.21274896..43422072..76964016.124223214.187527168
.0...0.414912.6654048.30938640.85654320.181790352.330218544.541990896..........

Empirical
k=1: a(n) = n^3
k=2: a(n) = 6*n^3 - 6*n^2
k=3: a(n) = 30*n^3 - 60*n^2 + 24*n for n>1
k=4: a(n) = 150*n^3 - 426*n^2 + 312*n - 48 for n>2
k=5: a(n) = 726*n^3 - 2640*n^2 + 2688*n - 720 for n>3
k=6: a(n) = 3534*n^3 - 15366*n^2 + 19536*n - 7056 for n>4
k=7: a(n) = 16926*n^3 - 85380*n^2 + 128832*n - 57312 for n>5
k=8: a(n) = 81390*n^3 - 463074*n^2 + 801216*n - 418032 for n>6
k=9: a(n) = 387966*n^3 - 2452704*n^2 + 4766544*n - 2833872 for n>7
k=10: a(n) = 1853886*n^3 - 12825630*n^2 + 27515184*n - 18252624 for n>8

The coefficient of n^3 is http://oeis.org/A001412 Number of [k-step] 
self-avoiding walks on cubic lattice



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