A076874 -id:A076874 - cp's OEIS frontend

A078526 Probability P(n) of the occurrence of a 2D self-trapping walk of length n.

1, 5, 31, 173, 1521, 4224, 33418, 184183, 1370009, 3798472, 26957026, 150399317, 1034714947, 2897704261, 19494273755, 109619578524, 724456628891

Offset: 7

Hugo Pfoertner, Nov 27 2002

This is a cleaner representation than the one given by A077483 and A077484, using the upper bound for the denominator A077484 given in A076874.

			See under A077483; the inclusion of a(7)=1 is somewhat artificial due to the occurrence of 2^(-1) in the denominator: P(7)=a(7)/(3^6 *2^(7-floor(sqrt(29))-3))= 1/(729*2^(7-5-3))=1/(729*2*(-1))=2/729 See also: "Count self-trapping walks up to length 23" provided at given link.

See under A077483

Hugo Pfoertner, Results for the 2D Self-Trapping Random Walk

Cf. A077483, A077484, A076874, A001411.

Fortran
```
c Program provided at given link
```

P(n) = a(n) / ( 3^(n-1) * 2^(n-floor((4*n+1)^(1/2))-3) ) = a(n) / ( 3^(n-1) * 2^(A076874(n)-3) )

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

Original entry on oeis.org

0, 1, 9, 7, 3, 36, 26, 13, 1, 100, 54, 19, 7, 247, 147, 68, 27, 12, 552, 294, 151

Offset: 7

Hugo Pfoertner, Nov 27 2002

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).

			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.

Hugo Pfoertner, Results for the 2D Self-Trapping Random Walk

Cf. A077482, A076874, A078528, A001411.

Fortran
```
c Program provided at given link
```

a(24)-a(27) from Sean A. Irvine, Jul 04 2025

cp's OEIS Frontend

A078526 Probability P(n) of the occurrence of a 2D self-trapping walk of length n.

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