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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 2 or 3; a(n) is the number of cells after n iterations.
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%I #16 Sep 08 2022 08:46:19

%S 1,20,352,6080,104704,1802240,31019008,533872640,9188540416,

%T 158144921600,2721848492032,46846013603840,806271544459264,

%U 13876822236200960,238835410589974528,4110620744461844480,70748315180918112256,1217656507884193710080,20957211028999804813312

%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 2 or 3; a(n) is the number of cells after n iterations.

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

%H G. C. Greubel, <a href="/A285393/b285393.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="/A285393/a285393.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) = 20, a(n) = 20*a(n-1) - 48*a(n-2).

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

%F a(n) = ((13 - 5*sqrt(13))*(10 - 2*sqrt(13))^n + (2*(5 + sqrt(13)))^n*(13 + 5*sqrt(13)))/26.

%F a(n) = (4*sqrt(3))^n * ChebyshevU(n, 5/(2*sqrt(3))). - _G. C. Greubel_, Dec 11 2021

%t LinearRecurrence[{20, -48}, {1, 20}, 19]

%o (Magma) [n le 2 select (20)^(n-1) else 20*Self(n-1) - 48*Self(n-2): n in [1..31]]; // _G. C. Greubel_, Dec 11 2021

%o (Sage) [(4*sqrt(3))^n * chebyshev_U(n, 5/(2*sqrt(3))) for n in (0..30)] # _G. C. Greubel_, Dec 11 2021

%Y Cf. A285391, A285392, A285394, A285395, A285396, A285397, A285398, A285399, A285400.

%K nonn

%O 0,2

%A _Peter Karpov_, Apr 19 2017