A285398 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; a(n) is the number of cells after n iterations.

1, 19, 452, 10948, 266300, 6484372, 157936172, 3847025764, 93707895260, 2282596837492, 55601016789068, 1354367059315396, 32990588541122684, 803607076375862356, 19574804963320797548, 476816346057854861860, 11614615234500986326556, 282916657894827156657460

Offset: 0

Author

Peter Karpov, Apr 23 2017

Keywords

• nonn
• easy
• nice

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
• Peter Karpov, Illustration of initial terms (n = 1..4)
• Index entries for linear recurrences with constant coefficients, signature (32,-195,216).

Crossrefs

Cf. A007482, A026597, A285391, A285392, A285393, A285394, A285395, A285396, A285397, A285399, A285400.

Programs

Magma

I:=[19, 452, 10948]; [1] cat [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

{1}~Join~LinearRecurrence[{32, -195, 216}, {19, 452, 10948}, 17]

PARI

Vec((1 - x)*(1 - 3*x)*(1 - 9*x) / (1 - 32*x + 195*x^2 - 216*x^3) + O(x^20)) \\ Colin Barker, Apr 23 2017

Sage

def A285398_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-13*x+39*x^2-27*x^3)/(1-32*x+195*x^2-216*x^3) ).list()
A285398_list(40) # G. C. Greubel, Dec 09 2021

Formula

a(0) = 1, a(1) = 19, a(2) = 452, a(3) = 10948, a(n) = 28*a(n-1) - 195*a(n-2) + 216*a(n-3).

G.f.: (1-13*x+39*x^2-27*x^3)/(1-32*x+195*x^2-216*x^3).