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

[Ron Hardin]

Adding 1-dimensional walks, and summarizing for dimensions 1-4

Formulas for n-step self-avoiding walks on a k^d d-cube summed over all starting 
positions for d=1..4

Numerology:
1.  They're polynomials in k of degree d, valid for k>n-2.
2.  The coefficient of k^d is the number of unrestricted self-avoiding walks of 
length n in d dimensions.
3.  The coefficient of k^(d-1) for n=2 is -2*d
4.  The coefficient of k^(d-1) for n=3 is -(8*d^2-4*d)

T(n,k)=Number of n-step self-avoiding walks on a k-long line summed over all 
starting positions
Empirical: T(1,k) = k
Empirical: T(2,k) = 2*k - 2
Empirical: T(3,k) = 2*k - 4 for k>1
Empirical: T(4,k) = 2*k - 6 for k>2
Empirical: T(5,k) = 2*k - 8 for k>3
Empirical: T(6,k) = 2*k - 10 for k>4
Empirical: T(7,k) = 2*k - 12 for k>5
Empirical: T(8,k) = 2*k - 14 for k>6

T(n,k)=Number of n-step self-avoiding walks on a kXk square summed over all 
starting positions
Empirical: T(1,k) = k^2
Empirical: T(2,k) = 4*k^2 - 4*k
Empirical: T(3,k) = 12*k^2 - 24*k + 8 for k>1
Empirical: T(4,k) = 36*k^2 - 100*k + 56 for k>2
Empirical: T(5,k) = 100*k^2 - 360*k + 272 for k>3
Empirical: T(6,k) = 284*k^2 - 1228*k + 1152 for k>4
Empirical: T(7,k) = 780*k^2 - 3960*k + 4432 for k>5
Empirical: T(8,k) = 2172*k^2 - 12500*k + 16096 for k>6

T(n,k)=Number of n-step self-avoiding walks on a kXkXk cube summed over all 
starting positions
Empirical: a(1,k) = k^3
Empirical: a(2,k) = 6*k^3 - 6*k^2
Empirical: a(3,k) = 30*k^3 - 60*k^2 + 24*k for k>1
Empirical: a(4,k) = 150*k^3 - 426*k^2 + 312*k - 48 for k>2
Empirical: a(5,k) = 726*k^3 - 2640*k^2 + 2688*k - 720 for k>3
Empirical: a(6,k) = 3534*k^3 - 15366*k^2 + 19536*k - 7056 for k>4
Empirical: a(7,k) = 16926*k^3 - 85380*k^2 + 128832*k - 57312 for k>5
Empirical: a(8,k) = 81390*k^3 - 463074*k^2 + 801216*k - 418032 for k>6
Empirical: a(9,k) = 387966*k^3 - 2452704*k^2 + 4766544*k - 2833872 for k>7
Empirical: a(10,k) = 1853886*k^3 - 12825630*k^2 + 27515184*k - 18252624 for k>8

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


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