A285391 - OEIS

%I #20 Sep 08 2022 08:46:19

%S 1,8,60,444,3276,24156,178092,1312956,9679500,71360028,526086252,

%T 3878455932,28593068364,210796144092,1554048476460,11456882559036,

%U 84463361313804,622687661115804,4590628614276588,33843405595099644,249503106984577740,1839407095720003932

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

%C Cell configuration converges to a fractal carpet with dimension 1.818...

%H Colin Barker, <a href="/A285391/b285391.txt">Table of n, a(n) for n = 0..1000</a>

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

%H Peter Karpov, <a href="/A285391/a285391.png">Illustration of cell configuration after 5 iterations</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (9,-12).

%F a(0) = 1, a(1) = 8, a(n) = 9*a(n-1) - 12*a(n-2).

%F G.f.: (1-x)/(1-9*x+12*x^2).

%F a(n) = (2^(-1-n)*((9-sqrt(33))^n*(-7+sqrt(33)) + (7+sqrt(33))*(9+sqrt(33))^n)) / sqrt(33). - _Colin Barker_, Apr 18 2017

%F a(n) = (2*sqrt(3))^(n-1)*( 2*sqrt(3)*ChebyshevU(n, 9/(4*sqrt(3))) - ChebyshevU(n-1, 9/(4*sqrt(3))) ). - _G. C. Greubel_, Dec 11 2021

%t LinearRecurrence[{9, -12}, {1, 8}, 16]

%o (PARI) Vec((1 - x) / (1 - 9*x + 12*x^2) + O(x^30)) \\ _Colin Barker_, Apr 18 2017

%o (Magma) [n le 2 select 8^(n-1) else 9*Self(n-1) - 12*Self(n-2): n in [1..31]]; // _G. C. Greubel_, Dec 11 2021

%o (Sage) [(2*sqrt(3))^(n-1)*( 2*sqrt(3)*chebyshev_U(n, 9/(4*sqrt(3))) - chebyshev_U(n-1, 9/(4*sqrt(3))) ) for n in (0..30)] # _G. C. Greubel_, Dec 11 2021

%Y Cf. A285392, A285393, A285394, A285395, A285396, A285397, A285398, A285399, A285400.

%K nonn,easy

%O 0,2

%A _Peter Karpov_, Apr 18 2017

%E More terms from _Colin Barker_, Apr 18 2017