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A285392 Start with a single cell at coordinates (0, 0), then iteratively subdivide the grid into 3 X 3 cells and remove the cells whose sum of modulo 2 coordinates is 0; a(n) is the number of cells after n iterations.

%I A285392 #21 Sep 08 2022 08:46:19
%S A285392 1,5,36,264,1944,14328,105624,778680,5740632,42321528,312006168,
%T A285392 2300197176,16957700568,125016939000,921660044184,6794737129656,
%U A285392 50092713636696,369297577174392,2722565630929176,20071519752269880,147972890199278808,1090897774766270712
%N A285392 Start with a single cell at coordinates (0, 0), then iteratively subdivide the grid into 3 X 3 cells and remove the cells whose sum of modulo 2 coordinates is 0; a(n) is the number of cells after n iterations.
%C A285392 Cell configuration converges to a fractal carpet with dimension 1.818...
%H A285392 Colin Barker, <a href="/A285392/b285392.txt">Table of n, a(n) for n = 0..1000</a>
%H A285392 Peter Karpov, <a href="http://inversed.ru/InvMem.htm#InvMem_26">InvMem, Item 26</a>
%H A285392 Peter Karpov, <a href="/A285392/a285392.png">Illustration of cell configuration after 5 iterations</a>
%H A285392 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (9,-12).
%F A285392 a(0) = 1, a(1) = 5, a(2) = 36, a(n) = 9*a(n-1) - 12*a(n-2).
%F A285392 G.f.: (1-4*x+3*x^2)/(1-9*x+12*x^2).
%F A285392 a(n) = (2^(-3-n)*((9-sqrt(33))^n*(-13+3*sqrt(33)) + (9+sqrt(33))^n*(13+3*sqrt(33)))) / sqrt(33) for n>0. - _Colin Barker_, Apr 18 2017
%F A285392 a(n) = (1/4)*([n=0] + (2*sqrt(3))^(n-1)*( 6*sqrt(3)*ChebyshevU(n, 9/(4*sqrt(3))) - 7*ChebyshevU(n-1, 9/(4*sqrt(3))) ) ). - _G. C. Greubel_, Dec 11 2021
%t A285392 {1}~Join~LinearRecurrence[{9, -12}, {5, 36}, 16]
%o A285392 (PARI) Vec((1 - x)*(1 - 3*x) / (1 - 9*x + 12*x^2) + O(x^30)) \\ _Colin Barker_, Apr 18 2017
%o A285392 (Magma) I:=[5, 36]; [1] cat [n le 2 select I[n] else 9*Self(n-1) - 12*Self(n-2): n in [1..31]]; // _G. C. Greubel_, Dec 11 2021
%o A285392 (Sage) [(1/4)*(bool(n==0) + (2*sqrt(3))^(n-1)*( 6*sqrt(3)*chebyshev_U(n, 9/(4*sqrt(3))) - 7*chebyshev_U(n-1, 9/(4*sqrt(3))) ) ) for n in (0..30)] # _G. C. Greubel_, Dec 11 2021
%Y A285392 Cf. A285391, A285393, A285394, A285395, A285396, A285397, A285398, A285399, A285400.
%K A285392 nonn,easy
%O A285392 0,2
%A A285392 _Peter Karpov_, Apr 18 2017
%E A285392 More terms from _Colin Barker_, Apr 18 2017