A078605
Sum of square displacements over all self-avoiding n-step walks on the cubic lattice with the first step specified. Numerator of mean square displacement s(n)=a(n)/(A001412(n)/6).
Original entry on oeis.org
1, 12, 97, 672, 4261, 25588, 147821, 830576, 4566917, 24692980, 131682825, 694386864, 3626770709, 18790632772, 96675376705, 494382431552, 2514666026897, 12730690730212, 64177763220925, 322314275563424, 1613192327878789, 8049191357609204, 40048773875769449, 198750753713937600
Offset: 1
a(2)=12 because the A001412(2)/6 = 5 different self-avoiding 2-step walks end at (1,0,-1), (1,0,1), (1,-1,0), (1,1,0)->d^2=2 and at (2,0,0)->d^2=4. a(2) = 4*2 + 1*4 = 12. See also "Distribution of end point distance" at first link.
A242355
Sum of squared end-to-end distances of all n-step self-avoiding walks on the 4-d cubic lattice.
Original entry on oeis.org
8, 128, 1416, 13568, 119960, 1009440, 8205656, 65068352, 506193144, 3879735776, 29378067080, 220265711040, 1637726387096, 12091336503584, 88727095777896, 647661676223168, 4705654523841704, 34049855885188128, 245482626441965048, 1764039730476165824
Offset: 1
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.
A359741
Number of n-step self-avoiding walks on a 3D cubic lattice whose end-to-end distance is an integer.
Original entry on oeis.org
1, 6, 6, 30, 78, 1134, 1350, 20574, 23238, 390606, 496998, 7614750, 10987926, 152120934, 237122526, 3110708214, 5017927638, 64718847438, 105210653478, 1362453235998
Offset: 0
a(3) = 30 as, in the first octant, there is one 3-step SAW whose end-to-end distance is an integer (1 unit):
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This can be walked in 24 different ways on a 3D cubic lattice. There are also the six walks directly along the x, y and z axes, giving a total of 24 + 6 = 30 walks.
A359133
Sum of square end-to-end displacements over all n-step self-avoiding walks of A359741.
Original entry on oeis.org
0, 6, 24, 78, 384, 8190, 8472, 178110, 193824, 4231662, 7072056, 102812142, 208526592, 2508914454, 5268441144, 62304671286, 124116667488, 1547651742990, 2850706506936, 38100453950670
Offset: 0
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