A329919 a(n) is the total number of squares after n iterations of the "Square Multiscale" substitution.

1, 17, 33, 49, 65, 321, 337, 849, 865, 1633, 1649, 2673, 6769, 6785, 8065, 20353, 20369, 21905, 46481, 46497, 48289, 89249, 154785, 154801, 156849, 218289, 480433, 480449, 482753, 568769, 1224129, 1224145, 1226705, 1341393, 2652113, 3700689, 3700705, 3703521

Offset: 0

Author

Felix Fröhlich, Nov 24 2019

Keywords

• nonn

Comments

The substitution starts with a single square. Then that square is subdivided into a "ring" of 16 small squares surrounding a larger square as shown in the example. In subsequent iterations, the same subdivision is applied to the largest square(s) present in that iteration.

Examples

			The basic subdivision rule:
  ----------------                     ----------------
  |              |                     |  |  |  |  |  |
  |              |                     ----------------
  |              |                     |  |        |  |
  |              |                     ----        ----
  |              |       ------>       |  |        |  |
  |              |                     ----        ----
  |              |                     |  |        |  |
  |              |                     ----------------
  |              |                     |  |  |  |  |  |
  ----------------                     ----------------
n = 1: The initial substitution subdivides the single square into 1 large and 16 small squares (as shown in the diagram above), so a(1) = 17.
n = 2, 3, 4: The largest square present after the previous iterations is the center square, so 16 new squares are added in each of those iterations. Thus, a(2) = a(1) + 16 = 33, a(3) = a(2) + 16 = 49, a(4) = a(3) + 16 = 65.
n = 5: This iteration subdivides the 16 outer squares (shown in the diagram above). 16^2 = 256, so a(5) = a(4) + 256 = 321.
		

Links

• Rémy Sigrist, PARI program for A329919
• Yotam Smilansky and Yaar Solomon, Multiscale Substitution Tilings, arXiv:2003.11735 [math.DS], 2020.
• Tilings Encyclopedia, Square Multiscale

Crossrefs

Cf. A328074, A329927.

Programs

PARI

See Links section.

Extensions

More terms from Rémy Sigrist, Nov 24 2019