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A285391
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.
10
1, 8, 60, 444, 3276, 24156, 178092, 1312956, 9679500, 71360028, 526086252, 3878455932, 28593068364, 210796144092, 1554048476460, 11456882559036, 84463361313804, 622687661115804, 4590628614276588, 33843405595099644, 249503106984577740, 1839407095720003932
OFFSET
0,2
COMMENTS
Cell configuration converges to a fractal carpet with dimension 1.818...
FORMULA
a(0) = 1, a(1) = 8, a(n) = 9*a(n-1) - 12*a(n-2).
G.f.: (1-x)/(1-9*x+12*x^2).
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
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
MATHEMATICA
LinearRecurrence[{9, -12}, {1, 8}, 16]
PROG
(PARI) Vec((1 - x) / (1 - 9*x + 12*x^2) + O(x^30)) \\ Colin Barker, Apr 18 2017
(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
(SageMath) [(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
KEYWORD
nonn,easy
AUTHOR
Peter Karpov, Apr 18 2017
EXTENSIONS
More terms from Colin Barker, Apr 18 2017
STATUS
approved