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

[Ron Hardin]

The same thing happens in two dimensions

Number of [k-step] self-avoiding walks on square lattice
The coefficients of n^2 below match http://oeis.org/A001411

Number of self-avoiding k-step walks on a nXn square grid summed over all 
starting positions

T(k,n) table starts
.1.4...9...16....25....36.....49.....64.....81....100....121.....144.....169
.0.8..24...48....80...120....168....224....288....360....440.....528.....624
.0.8..44..104...188...296....428....584....764....968...1196....1448....1724
.0.8..80..232...456...752...1120...1560...2072...2656...3312....4040....4840
.0.0.104..432...972..1712...2652...3792...5132...6672...8412...10352...12492
.0.0.128..800..2112..4008...6472...9504..13104..17272..22008...27312...33184
.0.0.112.1248..4152..8752..14932..22672..31972..42832..55252...69232...84772
.0.0.112.1976..8160.19312..35024..55104..79528.108296.141408..178864..220664
.0.0..40.2640.14520.39792..78168.128688.191068.265280.351324..449200..558908
.0.0...0.3696.26000.82032.175312.303328.464304.657848.883928.1142544.1433696

k=1: a(n) = n^2
k=2: a(n) = 4*n^2 - 4*n
k=3: a(n) = 12*n^2 - 24*n + 8 for n>1
k=4: a(n) = 36*n^2 - 100*n + 56 for n>2
k=5: a(n) = 100*n^2 - 360*n + 272 for n>3
k=6: a(n) = 284*n^2 - 1228*n + 1152 for n>4
k=7: a(n) = 780*n^2 - 3960*n + 4432 for n>5
k=8: a(n) = 2172*n^2 - 12500*n + 16096 for n>6
k=9: a(n) = 5916*n^2 - 38192*n + 55600 for n>7
k=10: a(n) = 16268*n^2 - 115548*n + 186528 for n>8


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





> 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)
> 
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