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

0,2

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.

Links

Table of n, a(n) for n=0..37.

Rémy Sigrist, PARI program for A329919

Yotam Smilansky and Yaar Solomon, Multiscale Substitution Tilings, arXiv:2003.11735 [math.DS], 2020.

Tilings Encyclopedia, Square Multiscale

Example

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.

Prog

(PARI) See Links section.

Crossrefs

Cf. A328074, A329927.

Sequence in context: A044443 A158057 A249356 * A346528 A286679 A116523

Adjacent sequences: A329916 A329917 A329918 * A329920 A329921 A329922

Keyword

nonn

Author

Felix Fröhlich, Nov 24 2019

Extensions

More terms from Rémy Sigrist, Nov 24 2019

Status

approved