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A009964 Powers of 20. 25

%I

%S 1,20,400,8000,160000,3200000,64000000,1280000000,25600000000,

%T 512000000000,10240000000000,204800000000000,4096000000000000,

%U 81920000000000000,1638400000000000000,32768000000000000000

%N Powers of 20.

%C 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

%C a(n) gives the number of small cubes in the n-th iteration of the Menger sponge fractal. - _Felix Fröhlich_, Jul 09 2016

%C Equivalently, the number of vertices in the n-Menger sponge graph.

%H T. D. Noe, <a href="/A009964/b009964.txt">Table of n, a(n) for n = 0..100</a>

%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MengerSponge.html">Menger Sponge</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MengerSpongeGraph.html">Menger Sponge Graph</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/VertexCount.html">Vertex Count</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Menger_sponge">Menger sponge</a>

%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (20).

%F G.f.: 1/(1-20*x).

%F E.g.f.: exp(20*x).

%F a(n) = A159991(n)/A000244(n). - _Reinhard Zumkeller_, May 02 2009

%F From _Vincenzo Librandi_, Nov 21 2010: (Start)

%F a(n) = 20^n.

%F a(n) = 20*a(n-1) for n > 0, a(0) = 1. (End)

%F a(n) = A000079(n)*A011557(n) = A000302(n)*A000351(n). - _Felix Fröhlich_, Jul 09 2016

%p [20^n$n=0..20]; # _Muniru A Asiru_, Nov 21 2018

%t 20^Range[0, 10] (* or *) LinearRecurrence[{20}, {1}, 20] (* _Eric W. Weisstein_, Aug 17 2017 *)

%o (Sage) [20^n for n in xrange(17)] # _Zerinvary Lajos_, Apr 29 2009

%o (MAGMA) [20^n: n in [0..100]] // _Vincenzo Librandi_, Nov 21 2010

%o (Maxima) makelist(20^n,n,0,30); /* _Martin Ettl_, Nov 05 2012 */

%o (PARI) a(n)=20^n \\ _Charles R Greathouse IV_, Jun 19 2015

%o (PARI) powers(20,12) \\ _Charles R Greathouse IV_, Jun 19 2015

%o (GAP) List([0..20],n->20^n); # _Muniru A Asiru_, Nov 21 2018

%o (Python) for n in range(0,20): print(20**n, end=', ') # _Stefano Spezia_, Nov 21 2018

%Y Cf. A291066 (edge count).

%Y Cf. A000079, A011557; A000302, A000351; A000244, A159991.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_

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Last modified November 20 12:13 EST 2019. Contains 329335 sequences. (Running on oeis4.)