A285393 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 2 or 3; a(n) is the number of cells after n iterations.

1, 20, 352, 6080, 104704, 1802240, 31019008, 533872640, 9188540416, 158144921600, 2721848492032, 46846013603840, 806271544459264, 13876822236200960, 238835410589974528, 4110620744461844480, 70748315180918112256, 1217656507884193710080, 20957211028999804813312

Offset: 0

Author

Peter Karpov, Apr 19 2017

Keywords

• nonn

Comments

Cell configuration converges to a fractal sponge 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).

Crossrefs

Cf. A285391, A285392, A285394, A285395, A285396, A285397, A285398, A285399, A285400.

Programs

Magma

[n le 2 select (20)^(n-1) else 20*Self(n-1) - 48*Self(n-2): n in [1..31]]; // G. C. Greubel, Dec 11 2021

Mathematica

LinearRecurrence[{20, -48}, {1, 20}, 19]

Sage

[(4*sqrt(3))^n * chebyshev_U(n, 5/(2*sqrt(3))) for n in (0..30)] # G. C. Greubel, Dec 11 2021

Formula

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

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

a(n) = ((13 - 5*sqrt(13))*(10 - 2*sqrt(13))^n + (2*(5 + sqrt(13)))^n*(13 + 5*sqrt(13)))/26.

a(n) = (4*sqrt(3))^n * ChebyshevU(n, 5/(2*sqrt(3))). - G. C. Greubel, Dec 11 2021