A293144 Number of vertices in a Menger Sponge constructed from a cubic lattice: a(n) = 20*a(n-1) - 24*A293143(n-1).

8, 64, 896, 15616, 295808, 5789440, 114790784, 2287878400, 45694209920, 913377753856, 18263504780672, 365237697021184, 7304494763023232, 146087821875273472, 2921739850525976960, 58434664314989709568, 1168692224736473884544

Offset: 0

Author

Steven Beard, Oct 01 2017

Keywords

• nonn
• easy

Comments

a(n) is the number of vertices of a (3^n+1)^3 cubic lattice minus the number of vertices missing for the openings within the sponge. The cubic honeycomb can be constructed by joining 20 cubes of the previous term and subtracting the overlapping vertices of 24 faces (see example).

Examples

			For a(0) we start with a simple cube, having a(0) = 8 corners.
For a(1), the cube is subdivided into 27 smaller cubes forming a lattice of 64 vertices. 7 cubes are removed (but the cubes have no facial or internal vertices to remove until the next stage).
Twenty a(1) cubes, each with 64 vertices, are then combined to form the lattice for a(2). The overlapped vertices of 24 faces (each with 16 vertices) are removed. Thus a(2) = (20*64) - (24*16) = 1280 - 384 = 896. The faces of the cubes are the Sierpinski Carpet grid of A293143.
		

Links

• M. F. Hasler, Table of n, a(n) for n = 0..800 (Terms n = 1 .. 768 computed by Colin Barker.)
• Eric Weisstein's World of Mathematics, Menger Sponge.
• Wikipedia, Menger sponge.
• Index entries for linear recurrences with constant coefficients, signature (32,-275,724,-480).

Crossrefs

Cf. A293143, A034472.

Programs

Mathematica

CoefficientList[Series[8 (1 - 24 x + 131 x^2 - 156 x^3)/((1 - x) (1 - 3 x) (1 - 8 x) (1 - 20 x)), {x, 0, 15}], x] (* Michael De Vlieger, Oct 09 2017 *)

PARI

Vec(8*(1 - 24*x + 131*x^2 - 156*x^3) / ((1 - x)*(1 - 3*x)*(1 - 8*x)*(1 - 20*x)) + O(x^30)) \\ Colin Barker, Oct 09 2017

PARI

A293144(n)=(255+133*3^(n+1)+63*4^n*5^(n+1)+3553*8^(n-1))*64/11305 \\ M. F. Hasler, Oct 16 2017

Formula

From Colin Barker, Oct 02 2017, adjusted for initial a(0) = 8 by M. F. Hasler, Oct 16 2017: (Start)

G.f.: 8*(1 - 24*x + 131*x^2 - 156*x^3) / ((1 - x)*(1 - 3*x)*(1 - 8*x)*(1 - 20*x)).

a(n) = 64*(3/133 + 3^(1+n)/85 + 11*8^(n-1)/35 + 9*20^n/323).

a(n) = 32*a(n-1) - 275*a(n-2) + 724*a(n-3) - 480*a(n-4) for n > 3.

(End)

a(n) = (64*(133*3^(n+1) + 63*4^n*5^(n+1) + 3553*8^(n-1) + 255)) / 11305.

Extensions

Edited to include initial term 8 by M. F. Hasler, Oct 16 2017