A009964 Powers of 20.

1, 20, 400, 8000, 160000, 3200000, 64000000, 1280000000, 25600000000, 512000000000, 10240000000000, 204800000000000, 4096000000000000, 81920000000000000, 1638400000000000000, 32768000000000000000

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

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.

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.

Links

• T. D. Noe, Table of n, a(n) for n = 0..100
• Allan Bickle, Degrees of Menger and Sierpinski Graphs, Congr. Num. 227 (2016) 197-208.
• Allan Bickle, MegaMenger Graphs, The College Mathematics Journal, 49 1 (2018) 20-26.
• Tanya Khovanova, Recursive Sequences
• Eric Weisstein's World of Mathematics, Menger Sponge
• Eric Weisstein's World of Mathematics, Menger Sponge Graph
• Eric Weisstein's World of Mathematics, Vertex Count
• Wikipedia, Menger sponge
• Index entries for linear recurrences with constant coefficients, signature (20).

Crossrefs

Cf. A291066 (edge count).

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

Cf. A291066, A083233, and A332705 on the surface area of the n-Menger sponge graph.

Programs

GAP

List([0..20],n->20^n); # Muniru A Asiru, Nov 21 2018

Magma

[20^n: n in [0..100]] // Vincenzo Librandi, Nov 21 2010

Maple

[20^n$n=0..20]; # Muniru A Asiru, Nov 21 2018

Mathematica

20^Range[0, 10] (* or *) LinearRecurrence[{20}, {1}, 20] (* Eric W. Weisstein, Aug 17 2017 *)

Maxima

makelist(20^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

Python

[20**n for n in range(21)] # Stefano Spezia, Nov 21 2018

Sage

[20^n for n in range(21)] # Zerinvary Lajos, Apr 29 2009

Formula

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) for n > 0, a(0) = 1. (End)

a(n) = A000079(n)*A011557(n) = A000302(n)*A000351(n). - Felix Fröhlich, Jul 09 2016