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 A293144 Number of vertices in a Menger Sponge constructed from a cubic lattice: a(n) = 20*a(n-1) - 24*A293143(n-1). 2
 8, 64, 896, 15616, 295808, 5789440, 114790784, 2287878400, 45694209920, 913377753856, 18263504780672, 365237697021184, 7304494763023232, 146087821875273472, 2921739850525976960, 58434664314989709568, 1168692224736473884544 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 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). 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). 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. EXAMPLE 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. 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 *) PROG (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 CROSSREFS Cf. A293143, A034472. Sequence in context: A133054 A098560 A087138 * A349266 A111984 A352721 Adjacent sequences:  A293141 A293142 A293143 * A293145 A293146 A293147 KEYWORD nonn,easy AUTHOR Steven Beard, Oct 01 2017 EXTENSIONS Edited to include initial term 8 by M. F. Hasler, Oct 16 2017 STATUS approved

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Last modified October 3 22:17 EDT 2022. Contains 357237 sequences. (Running on oeis4.)