A323856
Sum of square displacements over all self-avoiding n-step walks on 4-d cubic lattice with first step specified, A242355(n)/8.
Original entry on oeis.org
1, 16, 177, 1696, 14995, 126180, 1025707, 8133544, 63274143, 484966972, 3672258385, 27533213880, 204715798387, 1511417062948, 11090886972237, 80957709527896, 588206815480213, 4256231985648516, 30685328305245631, 220504966309520728, 1579874958814261407
Offset: 1
a(1) = 1 is the square displacement of the fixed initial step.
a(2) = 16, because one of the A010575(2)/8 = 7 end points is (2,0,0,0) with square distance 4 and the other 6 end points (1,-1,0,0), (1,1,0,0), (1,0,-1,0), (1,0,1,0), (1,0,0,-1), (1,0,0,1) all have square distance 2. 16 = 1*4 + 6*2.
a(3) = 177, because there are 6 end points with square distance 1, e.g., (0,1,0,0), 24 end points with square distance 3, e.g., (1,1,1,0), 18 end points with square distance 5, e.g., (2,1,0,0), and 1 end point with square distance 9, (3,0,0,0). 177 = 6*1 + 24*3 + 18*5 + 1*9.
A323857
Sum of end-to-end Manhattan distances over all self-avoiding n-step walks on 4-d cubic lattice.
Original entry on oeis.org
1, 14, 135, 1144, 9083, 69690, 522781, 3864524, 28243251, 204687550, 1473038447, 10542725976, 75096139471, 532846305962, 3767808141891, 26566180648012, 186826646453453
Offset: 1
a(3) = 135, because there are 6 (of A010575(3)/8=49) end points with Manhattan distance 1, (0,-1,0,0), (0,1,0,0), (0,0,-1,0), (0,0,1,0), (0,0,0,-1), (0,0,0,1), and the remaining 43 end points all have Manhattan distance 3, e.g., (3,0,0,0), (2,-1,0,0), ..., (1,-1,-1,0), ... 135 = 6*1 + 43*3.
A374397
a(n) is the number of 4-step self avoiding walks in the n-dimensional hypercubic lattice that start at the origin.
Original entry on oeis.org
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
- N. Madras and G. Slade, "The Self Avoiding Walk", Birkhäuser, 2013.
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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 *)
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