cp's OEIS Frontend

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

%I A329919 #23 Aug 21 2022 06:14:55
%S A329919 1,17,33,49,65,321,337,849,865,1633,1649,2673,6769,6785,8065,20353,
%T A329919 20369,21905,46481,46497,48289,89249,154785,154801,156849,218289,
%U A329919 480433,480449,482753,568769,1224129,1224145,1226705,1341393,2652113,3700689,3700705,3703521
%N A329919 a(n) is the total number of squares after n iterations of the "Square Multiscale" substitution.
%C A329919 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.
%H A329919 Rémy Sigrist, <a href="/A329919/a329919.gp.txt">PARI program for A329919</a>
%H A329919 Yotam Smilansky and Yaar Solomon, <a href="https://arxiv.org/abs/2003.11735">Multiscale Substitution Tilings</a>, arXiv:2003.11735 [math.DS], 2020.
%H A329919 Tilings Encyclopedia, <a href="https://tilings.math.uni-bielefeld.de/substitution/square-multiscale/">Square Multiscale</a>
%e A329919 The basic subdivision rule:
%e A329919   ----------------                     ----------------
%e A329919   |              |                     |  |  |  |  |  |
%e A329919   |              |                     ----------------
%e A329919   |              |                     |  |        |  |
%e A329919   |              |                     ----        ----
%e A329919   |              |       ------>       |  |        |  |
%e A329919   |              |                     ----        ----
%e A329919   |              |                     |  |        |  |
%e A329919   |              |                     ----------------
%e A329919   |              |                     |  |  |  |  |  |
%e A329919   ----------------                     ----------------
%e A329919 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.
%e A329919 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.
%e A329919 n = 5: This iteration subdivides the 16 outer squares (shown in the diagram above). 16^2 = 256, so a(5) = a(4) + 256 = 321.
%o A329919 (PARI) See Links section.
%Y A329919 Cf. A328074, A329927.
%K A329919 nonn
%O A329919 0,2
%A A329919 _Felix Fröhlich_, Nov 24 2019
%E A329919 More terms from _Rémy Sigrist_, Nov 24 2019