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Sum of square displacements over all self-avoiding n-step walks on 4-d cubic lattice with first step specified, A242355(n)/8.
3

%I #14 Feb 11 2019 02:05:48

%S 1,16,177,1696,14995,126180,1025707,8133544,63274143,484966972,

%T 3672258385,27533213880,204715798387,1511417062948,11090886972237,

%U 80957709527896,588206815480213,4256231985648516,30685328305245631,220504966309520728,1579874958814261407

%N Sum of square displacements over all self-avoiding n-step walks on 4-d cubic lattice with first step specified, A242355(n)/8.

%D For references and links see A010575 and A242355.

%e a(1) = 1 is the square displacement of the fixed initial step.

%e 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.

%e 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.

%Y Cf. A010575, A078605, A118313, A242355, A323857.

%K nonn,walk

%O 1,2

%A _Hugo Pfoertner_, Feb 03 2019