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%I #79 Jul 12 2023 12:31:26
%S 1,20,400,8000,160000,3200000,64000000,1280000000,25600000000,
%T 512000000000,10240000000000,204800000000000,4096000000000000,
%U 81920000000000000,1638400000000000000,32768000000000000000
%N Powers of 20.
%C Same as Pisot sequences E(1, 20), L(1, 20), P(1, 20), T(1, 20). Essentially same as Pisot sequences E(20, 400), L(20, 400), P(20, 400), T(20, 400). See A008776 for definitions of Pisot sequences.
%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 Allan Bickle, <a href="https://allanbickle.files.wordpress.com/2016/05/mengerspongedegree.pdf">Degrees of Menger and Sierpinski Graphs</a>, Congr. Num. 227 (2016) 197-208.
%H Allan Bickle, <a href="https://allanbickle.files.wordpress.com/2016/05/mengerspongeshort.pdf">MegaMenger Graphs</a>, The College Mathematics Journal, 49 1 (2018) 20-26.
%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 range(21)] # _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) [20**n for n in range(21)] # _Stefano Spezia_, Nov 21 2018
%Y Cf. A291066 (edge count).
%Y Cf. A000079, A011557; A000302, A000351; A000244, A159991.
%Y Cf. A291066, A083233, and A332705 on the surface area of the n-Menger sponge graph.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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