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A329919
a(n) is the total number of squares after n iterations of the "Square Multiscale" substitution.
5
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
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
Sequence in context: A044443 A158057 A249356 * A346528 A286679 A116523
KEYWORD
nonn
AUTHOR
Felix Fröhlich, Nov 24 2019
EXTENSIONS
More terms from Rémy Sigrist, Nov 24 2019
STATUS
approved