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A293143
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Number of vertex points in a Sierpinski Carpet grid subdivided into squares: a(n+1) = 8*a(n) - 8*(3^n+1), a(0) = 4.
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4
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4, 16, 96, 688, 5280, 41584, 330720, 2639920, 21101856, 168762352, 1349941344, 10799058352, 86391049632, 691124145520, 5528980409568, 44231805012784, 353854325311008, 2830834258114288, 22646673031792992, 181173381154980016
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OFFSET
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0,1
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COMMENTS
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Figurate number sequence for the Sierpinski Carpet lattice. See the faces of the cubes in "Image 2" in the Wikipedia link of A293144 for an example of the construction grid of the Sierpinski Carpet.
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LINKS
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Colin Barker, Table of n, a(n) for n = 0..1000 (Recomputed by M. F. Hasler to include the initial term 4.)
Eric Weisstein's World of Mathematics, Sierpinski Carpet.
Wikipedia, Sierpinski carpet.
Index entries for linear recurrences with constant coefficients, signature (12,-35,24).
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FORMULA
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From Colin Barker, Oct 02 2017, corrected for a(0) = 4 by M. F. Hasler, Oct 16 2017: (Start)
G.f.: 4*(1 - 8*x + 11*x^2) / ((1 - x)*(1 - 3*x)*(1 - 8*x)).
a(n) = 8*(5 + 11*2^(3*n-1) + 7*3^n) / 35.
a(n) = 12*a(n-1) - 35*a(n-2) + 24*a(n-3) for n > 2. (End)
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EXAMPLE
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The carpet is formed by squares within a square grid. The initial term is a(0) = 4 for the corners of the unit square. For n = 1 there are 3 X 3 squares, the middle one being open (empty), with 16 vertex points. At the next stage of recursion, these become eight squares with open center, now based on a square of 10 X 10 points. The remaining center square is empty, missing 4 points, thus the next term is 100 - 4 = 96 for a(2). In the next stage there are 8 squares missing 4 points and the new center is missing 64, thus the 28 square grid now has 784 - 32 - 64 = 688 for a(3). This carpet sequence becomes the faces for the cubes in the Menger Sponge recursion of A293144.
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MATHEMATICA
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FoldList[8 #1 - 8 (3^(#2-1) + 1) &, 4, Range@ 18] (* Michael De Vlieger, Oct 02 2017 *)
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PROG
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(PARI) prev=4; concat(prev, vector(20, n, prev=8*prev-8*(3^(n-1)+1))) \\ Colin Barker, Oct 08 2017
(PARI) Vec(4*(1 - 8*x + 11*x^2)/((1 - x)*(1 - 3*x)*(1 - 8*x)) + O(x^30)) \\ Colin Barker, Oct 09 2017
(PARI) A293143(n)=8*(5+11*2^(3*n-1)+7*3^n)/35 \\ M. F. Hasler, Oct 16 2017
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CROSSREFS
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Cf. A293144, A034472.
Sequence in context: A098580 A317998 A027745 * A032184 A130683 A111976
Adjacent sequences: A293140 A293141 A293142 * A293144 A293145 A293146
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KEYWORD
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nonn,easy
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AUTHOR
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Steven Beard, Oct 01 2017
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EXTENSIONS
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Edited to start with a(0) = 4 by M. F. Hasler, Oct 16 2017
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STATUS
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approved
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