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Number of 2-dimensional closed-loop self-avoiding paths on a square lattice where each path consists of steps with incrementing length from 1 to n.
9

%I #21 May 14 2020 16:03:22

%S 0,0,0,0,0,0,8,24,0,0,40,112,0,0,1376,2008,0,0,21720,60848,0,0,635544,

%T 1517368,0,0,20008456,46010640,0,0,640819936,1571759136,0,0,

%U 22704325648,55436103264

%N Number of 2-dimensional closed-loop self-avoiding paths on a square lattice where each path consists of steps with incrementing length from 1 to n.

%C This sequence gives the number of closed-loop self avoiding walks on a 2D square lattice where the walk starts with a step length of 1 which then increments by 1 after each step up until the step length is n. No closed-loop path is possible until n = 7.

%C Like A010566 all possible paths are counted, including those that are equivalent via rotation and reflection.

%H A. J. Guttmann and I. G. Enting, <a href="https://doi.org/10.1088/0305-4470/21/3/009">The size and number of rings on the square lattice</a>, J. Phys. A 21 (1988), L165-L172.

%H Scott R. Shannon, <a href="/A334720/a334720.txt">Images of the closed-loops for n=7,8,11,12,15</a>.

%e a(1) to a(6) = 0 as no closed-loop is possible.

%e a(7) = 8 as there is one path which forms a closed loop which can be walked in 8 different ways on a 2D square lattice. The path is:

%e .

%e 5

%e *---.---.---.---.---*

%e | |

%e . .

%e | |

%e . . 4

%e | |

%e 6 . .

%e | | 3

%e . *---.---.---*

%e | |

%e . . 2

%e | |

%e *---.---.---.---.---.---.---X---*

%e 7 1

%e .

%e See the attached link for text images of the closed loops for other n values.

%Y Cf. A010566, A334877, A002931, A334756.

%K nonn,more,walk

%O 1,7

%A _Scott R. Shannon_, May 08 2020