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%I #22 Jan 02 2023 12:30:54
%S 0,4,32,156,608,2116,6816,20844,61376,175628,491248,1349172,3650144,
%T 9751532,25774672,67501556,175375136,452454276,1160098576,2958123556,
%U 7505767840,18959922796,47701159264,119570463980,298719578688,743984084700,1847709517360,4576818079076,11309417827072
%N Sum of square end-to-end distance over all self-avoiding n-step walks on a square lattice where no adjacent points are allowed, except those for consecutive steps.
%C The corresponding number of n-step walks is given in A173380.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes the sequence A173380).
%H Sequence Fans Mailing list, <a href="http://list.seqfan.eu/oldermail/seqfan/2010-November/006470.html">discussion of the sequence A173380</a>, November 2010.
%e The allowed 4-step walks with their associated end-to-end square distances are:
%e .
%e + 10
%e 4 | 8 8 8 16
%e +--+ + +--+ + + X--+---+---+---+
%e | | | 10 | |
%e + + + +--+--+ +--+ + +--+ 10 + 10
%e | | | | | | | |
%e X--+ X--+ X--+ X--+ X--+ X--+--+ X--+--+ X--+--+--+
%e .
%e The eight non-straight walks sum to 68, and these can be walked in eight ways on the square lattice. The remaining straight walk can be walking in four ways. Thus a(4) = 68 * 8 + 16 * 4 = 608.
%Y Cf. A173380, A001411.
%K nonn,walk
%O 0,2
%A _Scott R. Shannon_, Aug 25 2020
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