A328074 Coordination sequence for a certain multiscale substitution tiling of the plane by squares.

1, 12, 16, 16, 40, 52, 96, 84, 72, 92, 128, 104, 68, 104, 112, 148, 168, 140, 136, 248, 208, 264, 264, 284, 264, 364, 384, 412, 328, 404, 400, 496, 392, 408, 416, 424, 372, 408, 456, 468, 468, 504, 540, 576, 572, 616, 608, 616, 576, 616, 556, 576, 620, 612

Offset: 0

Author

N. J. A. Sloane, Oct 07 2019, based on an email message from Yotam Smilansky

Keywords

• nonn

Comments

This substitution rule dissects the unit square into a central square of side 3/5 and 16 surrounding squares of side 1/5.

What is the limiting shape of the contours (if it exists)?

From Lars Blomberg, Oct 18 2019: (Start)

Let s be the size of a square. The substitution rule is to replace it by one central square (size s*3/5) and sixteen smaller squares around it (size s*1/5).

Start with a single square as generation 0.

For each new generation first substitute the central square, let c be the size of the new central square.

Then substitute all non-central squares with size >= c. Repeat the last step if required. (End)

Links

• Lars Blomberg, Table of n, a(n) for n = 0..596
• Lars Blomberg, Illustration of coordination sequence for generation 12
• Yotam Smilansky, Patterns and Partitions, Experimental Mathematics Seminar, Rutgers University, Oct 03 2019.
• Yotam Smilansky, Central portion of the tiling.
• Yotam Smilansky, Colored picture of central portion of tiling showing contours.
• Yotam Smilansky, Yaar Solomon, Multiscale Substitution Tilings, arXiv:2003.11735 [math.DS], 2020.

Extensions

More terms from Lars Blomberg, Oct 18 2019