A285394 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 0 or 1; a(n) is the number of cells after n iterations.

1, 7, 116, 1984, 34112, 587008, 10102784, 173879296, 2992652288, 51506839552, 886489481216, 15257461325824, 262597731418112, 4519596484722688, 77787238586384384, 1338804140460998656, 23042295357073522688, 396583308399342518272, 6825635990847321276416

Offset

0,2

Comments

Cell configuration converges to a fractal with dimension 2.590...

Links

G. C. Greubel, Table of n, a(n) for n = 0..750

Peter Karpov, InvMem, Item 26

Peter Karpov, Illustration of initial terms (n = 1..4)

Index entries for linear recurrences with constant coefficients, signature (20,-48).

Formula

a(0) = 1, a(1) = 7, a(2) = 116, a(n) = 20*a(n-1) - 48*a(n-2).

G.f.: (1-13*x+24*x^2)/(1-20*x+48*x^2).

a(n) = (3*(10-2*sqrt(13))^n*(13+sqrt(13)) + (2*(5+sqrt(13)))^n*(91+23*sqrt(13)))/(52*(5+sqrt(13))) for n > 0.

a(n) = (1/2)*[n=0] + (4*sqrt(3))^(n-1)*(2*sqrt(3)*ChebyshevU(n, 5/(2*sqrt(3))) - 3*ChebyshevU(n-1, 5/(2*sqrt(3)))). - G. C. Greubel, Dec 10 2021

Mathematica

{1}~Join~LinearRecurrence[{20, -48}, {7, 116}, 18]
CoefficientList[Series[(1 - 13x + 24x^2)/(1 - 20x + 48x^2), {x, 0, 40}], x] (* Indranil Ghosh, Apr 19 2017 *)

Prog

(Magma) I:=[7, 116]; [n le 2 select I[n] else 20*Self(n-1) - 48*Self(n-2): n in [1..31]]; // G. C. Greubel, Dec 10 2021
(Sage) [(1/2)*bool(n==0) + (4*sqrt(3))^(n-1)*(2*sqrt(3)*chebyshev_U(n, 5/(2*sqrt(3))) - 3*chebyshev_U(n-1, 5/(2*sqrt(3)))) for n in (0..30)] # G. C. Greubel, Dec 10 2021

Crossrefs

Cf. A285391, A285392, A285393, A285395, A285396, A285397, A285398, A285399, A285400.

Sequence in context: A220181 A328813 A178297 * A339390 A376042 A251585

Adjacent sequences: A285391 A285392 A285393 * A285395 A285396 A285397

Keyword

nonn

Author

Peter Karpov, Apr 19 2017

Status

approved