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

0,2

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.

Crossrefs

Sequence in context: A135451 A175784 A143090 * A068394 A189685 A126763

Adjacent sequences: A328071 A328072 A328073 * A328075 A328076 A328077

Keyword

nonn

Author

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

Extensions

More terms from Lars Blomberg, Oct 18 2019

Status

approved