A285397 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 3; a(n) is the number of cells after n iterations.

1, 26, 646, 15818, 385822, 9401330, 229023958, 5578844858, 135894050926, 3310204057250, 80632220390758, 1964094376340522, 47842741143064894, 1165385872796078546, 28387257791866411894, 691476036231391881242, 16843441238514542846350, 410283940250387099210114

Offset: 0

Author

Peter Karpov, Apr 23 2017

Keywords

• nonn
• easy

Comments

Cell configuration converges to a fractal with dimension 2.906...

Links

• Colin Barker, Table of n, a(n) for n = 0..700
• Peter Karpov, InvMem, Item 26
• Index entries for linear recurrences with constant coefficients, signature (32,-195,216).

Crossrefs

Cf. A007482, A026597, A285391, A285392, A285393, A285394, A285395, A285396, A285398, A285399, A285400.

Programs

Magma

I:=[1, 26, 646]; [n le 3 select I[n] else 32*Self(n-1) - 195*Self(n-2) + 216*Self(n-3) : n in [1..41]]; // G. C. Greubel, Dec 09 2021

Mathematica

LinearRecurrence[{32, -195, 216}, {1, 26, 646}, 18]

PARI

Vec((1 - 3*x)^2 / (1 - 32*x + 195*x^2 - 216*x^3) + O(x^20)) \\ Colin Barker, Apr 23 2017

Sage

def A285397_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-6*x+9*x^2)/(1-32*x+195*x^2-216*x^3) ).list()
A285397_list(40) # G. C. Greubel, Dec 09 2021

Formula

a(0) = 1, a(1) = 26, a(2) = 646, a(n) = 28*a(n-1) - 195*a(n-2) + 216*a(n-3).

G.f.: (1-6*x+9*x^2)/(1-32*x+195*x^2-216*x^3).