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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.
2
1, 4, 4, 12, 28, 76, 164, 732, 1044, 4924, 6724, 30636, 43972, 190516, 313996, 1197908, 2284260, 7678188, 16257604, 50524252, 113052396, 341811828, 773714436, 2358452388, 5245994292, 16447462492, 35395532236, 115129727188, 238542983748, 804980005276
OFFSET
0,2
COMMENTS
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.
EXAMPLE
a(3) = 12 as, in the first quadrant, there is one 3-step SAW whose end-to-end distance is an integer (1 unit):
.
X---.
|
X---.
.
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.
KEYWORD
nonn,walk
AUTHOR
Scott R. Shannon, Jan 12 2023
STATUS
approved