A374397 a(n) is the number of 4-step self avoiding walks in the n-dimensional hypercubic lattice that start at the origin.

2, 100, 726, 2696, 7210, 15852, 30590, 53776, 88146, 136820, 203302, 291480, 405626, 550396, 730830, 952352, 1220770, 1542276, 1923446, 2371240, 2893002, 3496460, 4189726, 4981296, 5880050, 6895252, 8036550, 9313976, 10737946, 12319260, 14069102, 15999040, 18121026

Offset

1,1

Comments

We have the formula below because we have 2*n choices for the first step, and (2*n-1)^3 choices for the next three, but have counted exactly 2*n*(2*n-1)*(2*n-2) self-intersecting walks.

References

N. Madras and G. Slade, "The Self Avoiding Walk", Birkhäuser, 2013.

Links

Paolo Xausa, Table of n, a(n) for n = 1..10000

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Formula

a(n) = 16*n^4 - 24*n^3 + 8*n^2 + 2*n.

G.f.: 2*x*(1 + 45*x + 123*x^2 + 23*x^3)/(1 - x)^5. - Stefano Spezia, Jul 07 2024

Mathematica

A374397[n_] := 2*n*(4*n*(n - 1)*(2*n - 1) + 1);
Array[A374397, 50] (* or *)
LinearRecurrence[{5, -10, 10, -5, 1}, {2, 100, 726, 2696, 7210}, 50] (* Paolo Xausa, Sep 23 2024 *)

Crossrefs

Cf. A010575.

Sequence in context: A276386 A333023 A171396 * A202945 A230816 A174646

Adjacent sequences: A374394 A374395 A374396 * A374398 A374399 A374400

Keyword

nonn,walk,easy

Author

Johann Peters, Jul 07 2024

Status

approved