A337367 Sum of square end-to-end distance over all self-avoiding n-step walks on a square lattice where no adjacent points are allowed, except those for consecutive steps.

0, 4, 32, 156, 608, 2116, 6816, 20844, 61376, 175628, 491248, 1349172, 3650144, 9751532, 25774672, 67501556, 175375136, 452454276, 1160098576, 2958123556, 7505767840, 18959922796, 47701159264, 119570463980, 298719578688, 743984084700, 1847709517360, 4576818079076, 11309417827072

Offset

0,2

Comments

The corresponding number of n-step walks is given in A173380.

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes the sequence A173380).

Links

Table of n, a(n) for n=0..28.

Sequence Fans Mailing list, discussion of the sequence A173380, November 2010.

Example

The allowed 4-step walks with their associated end-to-end square distances are:
.
         + 10
4        |        8              8      8           16
+--+     +     +--+              +      +    X--+---+---+---+
   |     |     |          10     |      |
   +     +     +     +--+--+  +--+      +        +--+ 10      + 10
   |     |     |     |        |         |        |            |
X--+  X--+  X--+  X--+     X--+   X--+--+  X--+--+   X--+--+--+
.
The eight non-straight walks sum to 68, and these can be walked in eight ways on the square lattice. The remaining straight walk can be walking in four ways. Thus a(4) = 68 * 8 + 16 * 4 = 608.

Crossrefs

Cf. A173380, A001411.

Sequence in context: A302267 A113154 A302966 * A334323 A270161 A270977

Adjacent sequences: A337364 A337365 A337366 * A337368 A337369 A337370

Keyword

nonn,walk

Author

Scott R. Shannon, Aug 25 2020

Status

approved