How many different shapes can I make with this toy?

combinatorics discrete-mathematics recreational-mathematics combinatorial-geometry

I believe these are Self-Avoiding walks. These related to Sloane sequence A001411:

1, 4, 12, 36, 100, 284, 780, 2172, 5916,...


(from MathWorld - A Wolfram Web Resource: wolfram.com)

The self-avoiding walks on a cubic lattice, A001412

1, 6, 30, 150, 726, 3534, 16926, 81390,...

Sloane's Encyclopedia of Integer Sequences offers 317 number patterns related to self-avoiding walks.

Self-avoiding walks are related to computational chemistry and statistical mechanics.

Such problem are related to the work of 2010 Fields Medalist Stanislav Smirnov who showed the number of such paths on the hexagonal lattice grow was $(\sqrt{2 + \sqrt{2}})^n$

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