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A374397
a(n) is the number of 4-step self avoiding walks in the n-dimensional hypercubic lattice that start at the origin.
1
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.
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
KEYWORD
nonn,walk,easy
AUTHOR
Johann Peters, Jul 07 2024
STATUS
approved