

A078527


Number of maximally 2constrained walks on square lattice trapped after n steps.


2



0, 1, 9, 7, 3, 36, 26, 13, 1, 100, 54, 19, 7, 247, 147, 68, 27
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OFFSET

7,3


COMMENTS

In a 2D selfavoiding walk there may be steps, where the number of free target positions is less than 3. A step is called kconstrained, if only k<3 neighbors were not visited before. Selftrapping occurs at step n (the next step would have k=0). A maximally 2constrained nstep walk contains nfloor((4*n+1)^(1/2))2 steps with k=2 (conjectured). The first step is chosen fixed (0,0)>(1,0), all other steps have k=3. This sequence counts those walks among all possible selftrapping nstep walks A077482(n).


LINKS



EXAMPLE

a(7)=0 because the unique shortest possible selftrapping walk has no constrained steps. Of the A077482(10)=25 selftrapping walks of length n=10, there are A078528(10)=5 unconstrained walks (9 steps with free choice of direction). a(10)=7 walks are maximally 2constrained containing 2 steps with k=2. Among the remaining 13 walks there are 11 walks having 1 step with k=2 and 2 walks have 1 forced step k=1. An illustration of all unconstrained and all maximally 2constrained 10step walks is given in the first link under "5 Unconstrained and 7 maximally 2constrained walks of length 10". a(15)=1 is a unique ("perfectly constrained") walk visiting all lattice points of a 4*4 square, see "Examples for walks with the maximum number of constrained steps" provided at the given link.


PROG

FORTRAN program provided at given link


CROSSREFS



KEYWORD

more,nonn


AUTHOR



STATUS

approved



