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A078527 Number of maximally 2-constrained walks on square lattice trapped after n steps.

%I A078527 #11 Jul 04 2025 02:28:19
%S A078527 0,1,9,7,3,36,26,13,1,100,54,19,7,247,147,68,27,12,552,294,151
%N A078527 Number of maximally 2-constrained walks on square lattice trapped after n steps.
%C A078527 In a 2D self-avoiding walk there may be steps, where the number of free target positions is less than 3. A step is called k-constrained, if only k<3 neighbors were not visited before. Self-trapping occurs at step n (the next step would have k=0). A maximally 2-constrained n-step walk contains n-floor((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 self-trapping n-step walks A077482(n).
%H A078527 Hugo Pfoertner, <a href="http://www.randomwalk.de/stw2d.html">Results for the 2D Self-Trapping Random Walk</a>
%e A078527 a(7)=0 because the unique shortest possible self-trapping walk has no constrained steps. Of the A077482(10)=25 self-trapping 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 2-constrained 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 2-constrained 10-step walks is given in the first link under "5 Unconstrained and 7 maximally 2-constrained 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.
%o A078527 (Fortran) c Program provided at given link
%Y A078527 Cf. A077482, A076874, A078528, A001411.
%K A078527 more,nonn
%O A078527 7,3
%A A078527 _Hugo Pfoertner_, Nov 27 2002
%E A078527 a(24)-a(27) from _Sean A. Irvine_, Jul 04 2025