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1, 20, 400, 8000, 160000, 3200000, 64000000, 1280000000, 25600000000, 512000000000, 10240000000000, 204800000000000, 4096000000000000, 81920000000000000, 1638400000000000000, 32768000000000000000
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,2
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COMMENTS
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The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n >= 1, a(n) equals the number of 20-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011
a(n) gives the number of small cubes in the n-th iteration of the Menger sponge fractal. - Felix Fröhlich, Jul 09 2016
Equivalently, the number of vertices in the n-Menger sponge graph.
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LINKS
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T. D. Noe, Table of n, a(n) for n = 0..100
Tanya Khovanova, Recursive Sequences
Eric Weisstein's World of Mathematics, Menger Sponge
Eric Weisstein's World of Mathematics, Vertex Count
Wikipedia, Menger sponge
Index entries for linear recurrences with constant coefficients, signature (20).
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FORMULA
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G.f.: 1/(1-20*x).
E.g.f.: exp(20*x).
a(n) = A159991(n)/A000244(n). - Reinhard Zumkeller, May 02 2009
From Vincenzo Librandi, Nov 21 2010: (Start)
a(n) = 20^n.
a(n) = 20*a(n-1) n > 0 a(0)=1. (End)
a(n) = A000079(n)*A011557(n) = A000302(n)*A000351(n). - Felix Fröhlich, Jul 09 2016
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MATHEMATICA
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20^Range[0, 10] (* Eric W. Weisstein, Aug 17 2017 *)
LinearRecurrence[{20}, {1}, 20] (* Eric W. Weisstein, Aug 17 2017 *)
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PROG
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(Sage) [lucas_number1(n, 20, 0) for n in xrange(1, 17)] # Zerinvary Lajos, Apr 29 2009
(MAGMA)[20^n: n in [0..100]] // Vincenzo Librandi, Nov 21 2010
(Maxima) A009964(n):=20^n$
makelist(A009964(n), n, 0, 30); /* Martin Ettl, Nov 05 2012 */
(PARI) a(n)=20^n \\ Charles R Greathouse IV, Jun 19 2015
(PARI) powers(20, 12) \\ Charles R Greathouse IV, Jun 19 2015
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CROSSREFS
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Cf. A291066 (edge count).
Sequence in context: A224019 A171294 A267793 * A285874 A007577 A048987
Adjacent sequences: A009961 A009962 A009963 * A009965 A009966 A009967
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KEYWORD
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nonn,easy,changed
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AUTHOR
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N. J. A. Sloane
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STATUS
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approved
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