A285392 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 0; a(n) is the number of cells after n iterations.

1, 5, 36, 264, 1944, 14328, 105624, 778680, 5740632, 42321528, 312006168, 2300197176, 16957700568, 125016939000, 921660044184, 6794737129656, 50092713636696, 369297577174392, 2722565630929176, 20071519752269880, 147972890199278808, 1090897774766270712

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) = 5, a(2) = 36, a(n) = 9*a(n-1) - 12*a(n-2).

G.f.: (1-4*x+3*x^2)/(1-9*x+12*x^2).

a(n) = (2^(-3-n)*((9-sqrt(33))^n*(-13+3*sqrt(33)) + (9+sqrt(33))^n*(13+3*sqrt(33)))) / sqrt(33) for n>0. - Colin Barker, Apr 18 2017

a(n) = (1/4)*([n=0] + (2*sqrt(3))^(n-1)*( 6*sqrt(3)*ChebyshevU(n, 9/(4*sqrt(3))) - 7*ChebyshevU(n-1, 9/(4*sqrt(3))) ) ). - G. C. Greubel, Dec 11 2021

Mathematica

{1}~Join~LinearRecurrence[{9, -12}, {5, 36}, 16]

Prog

(PARI) Vec((1 - x)*(1 - 3*x) / (1 - 9*x + 12*x^2) + O(x^30)) \\ Colin Barker, Apr 18 2017
(Magma) I:=[5, 36]; [1] cat [n le 2 select I[n] else 9*Self(n-1) - 12*Self(n-2): n in [1..31]]; // G. C. Greubel, Dec 11 2021
(Sage) [(1/4)*(bool(n==0) + (2*sqrt(3))^(n-1)*( 6*sqrt(3)*chebyshev_U(n, 9/(4*sqrt(3))) - 7*chebyshev_U(n-1, 9/(4*sqrt(3))) ) ) for n in (0..30)] # G. C. Greubel, Dec 11 2021

Crossrefs

Cf. A285391, A285393, A285394, A285395, A285396, A285397, A285398, A285399, A285400.

Sequence in context: A055270 A297576 A164110 * A327091 A201351 A253470

Adjacent sequences: A285389 A285390 A285391 * A285393 A285394 A285395

Keyword

nonn,easy

Author

Peter Karpov, Apr 18 2017

Extensions

More terms from Colin Barker, Apr 18 2017

Status

approved