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Start with a single cell at coordinates (0, 0, 0), then iteratively subdivide the grid into 3 X 3 X 3 cells and remove the cells whose sum of modulo 2 coordinates is 0 or 1; a(n) is the number of cells after n iterations.
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%I #19 Sep 08 2022 08:46:19

%S 1,7,116,1984,34112,587008,10102784,173879296,2992652288,51506839552,

%T 886489481216,15257461325824,262597731418112,4519596484722688,

%U 77787238586384384,1338804140460998656,23042295357073522688,396583308399342518272,6825635990847321276416

%N Start with a single cell at coordinates (0, 0, 0), then iteratively subdivide the grid into 3 X 3 X 3 cells and remove the cells whose sum of modulo 2 coordinates is 0 or 1; a(n) is the number of cells after n iterations.

%C Cell configuration converges to a fractal with dimension 2.590...

%H G. C. Greubel, <a href="/A285394/b285394.txt">Table of n, a(n) for n = 0..750</a>

%H Peter Karpov, <a href="http://inversed.ru/InvMem.htm#InvMem_26">InvMem, Item 26</a>

%H Peter Karpov, <a href="/A285394/a285394.jpg">Illustration of initial terms (n = 1..4)</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (20,-48).

%F a(0) = 1, a(1) = 7, a(2) = 116, a(n) = 20*a(n-1) - 48*a(n-2).

%F G.f.: (1-13*x+24*x^2)/(1-20*x+48*x^2).

%F a(n) = (3*(10-2*sqrt(13))^n*(13+sqrt(13)) + (2*(5+sqrt(13)))^n*(91+23*sqrt(13)))/(52*(5+sqrt(13))) for n > 0.

%F a(n) = (1/2)*[n=0] + (4*sqrt(3))^(n-1)*(2*sqrt(3)*ChebyshevU(n, 5/(2*sqrt(3))) - 3*ChebyshevU(n-1, 5/(2*sqrt(3)))). - _G. C. Greubel_, Dec 10 2021

%t {1}~Join~LinearRecurrence[{20, -48}, {7, 116}, 18]

%t CoefficientList[Series[(1 - 13x + 24x^2)/(1 - 20x + 48x^2), {x, 0, 40}], x] (* _Indranil Ghosh_, Apr 19 2017 *)

%o (Magma) I:=[7,116]; [n le 2 select I[n] else 20*Self(n-1) - 48*Self(n-2): n in [1..31]]; // _G. C. Greubel_, Dec 10 2021

%o (Sage) [(1/2)*bool(n==0) + (4*sqrt(3))^(n-1)*(2*sqrt(3)*chebyshev_U(n, 5/(2*sqrt(3))) - 3*chebyshev_U(n-1, 5/(2*sqrt(3)))) for n in (0..30)] # _G. C. Greubel_, Dec 10 2021

%Y Cf. A285391, A285392, A285393, A285395, A285396, A285397, A285398, A285399, A285400.

%K nonn

%O 0,2

%A _Peter Karpov_, Apr 19 2017