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.

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...

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Peter Karpov, InvMem, Item 26

Peter Karpov, Illustration of cell configuration after 5 iterations

Index entries for linear recurrences with constant coefficients, signature (9,-12).

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
(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

Crossrefs

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

Sequence in context: A093132 A094169 A129325 * A001267 A099156 A245391

Adjacent sequences: A285388 A285389 A285390 * A285392 A285393 A285394

Keyword

nonn,easy

Author

Peter Karpov, Apr 18 2017

Extensions

More terms from Colin Barker, Apr 18 2017

Status

approved