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A359709 Number of n-step self-avoiding walks on a 2D square lattice whose end-to-end distance is an integer.

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%I A359709 #27 Jan 15 2023 15:11:35
%S A359709 1,4,4,12,28,76,164,732,1044,4924,6724,30636,43972,190516,313996,
%T A359709 1197908,2284260,7678188,16257604,50524252,113052396,341811828,
%U A359709 773714436,2358452388,5245994292,16447462492,35395532236,115129727188,238542983748,804980005276
%N A359709 Number of n-step self-avoiding walks on a 2D square lattice whose end-to-end distance is an integer.
%C A359709 The walks counted are all those directly along and x or y axes, and all walks whose final (|x|,|y|) lattice point are the two legs of a Pythagorean triple.
%H A359709 Wikipedia, <a href="https://en.wikipedia.org/wiki/Self-avoiding_walk">Self-avoiding walk</a>.
%e A359709 a(3) = 12 as, in the first quadrant, there is one 3-step SAW whose end-to-end distance is an integer (1 unit):
%e A359709 .
%e A359709      X---.
%e A359709          |
%e A359709      X---.
%e A359709 .
%e A359709 This can be walked in 8 different ways on a 2D square lattice. There are also the four walks directly along the x and y axes, giving a total of 8 + 4 = 12 walks.
%Y A359709 Cf. A359073, A103606, A359741, A001411, A356617, A173380, A337353, A358036, A358046.
%K A359709 nonn,walk
%O A359709 0,2
%A A359709 _Scott R. Shannon_, Jan 12 2023