A011557 Powers of 10: a(n) = 10^n.

1, 10, 100, 1000, 10000, 100000, 1000000, 10000000, 100000000, 1000000000, 10000000000, 100000000000, 1000000000000, 10000000000000, 100000000000000, 1000000000000000, 10000000000000000, 100000000000000000, 1000000000000000000

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

Same as Pisot sequences E(1, 10), L(1, 10), P(1, 10), T(1, 10). Essentially same as Pisot sequences E(10, 100), L(10, 100), P(10, 100), T(10, 100). See A008776 for definitions of Pisot sequences.

Same as k^n in base k. - Dominick Cancilla, Aug 02 2010 [Corrected by Jianing Song, Sep 17 2022]

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 10-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011

Smallest n+1 digit number greater than 0 (with offset 0). - Wesley Ivan Hurt, Jan 17 2014

Numbers with digit sum = 1, or, A007953(a(n)) = 1. - Reinhard Zumkeller, Jul 17 2014

Does not satisfy Benford's law. - N. J. A. Sloane, Feb 14 2017

References

• Philip Morrison et al., Powers of Ten, Scientific American Press, 1982 and later editions.
• S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

Links

• T. D. Noe, Table of n, a(n) for n = 0..100
• Kees Boeke, Cosmic View: The Universe in 40 Jumps (1957) [The original "powers of ten" book]
• P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
• Charles and Ray Eames, Powers of Ten
• Tanya Khovanova, Recursive Sequences
• Robert Price, Comments on A011557 concerning Elementary Cellular Automata, Feb 21 2016
• Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
• Science, Optics and You, Secret Worlds: The Universe Within [Powers of Ten]
• Eric Weisstein's World of Mathematics, 10
• Eric Weisstein's World of Mathematics, Digitaddition
• Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
• Wikipedia, Powers of Ten
• S. Wolfram, A New Kind of Science
• Index entries for linear recurrences with constant coefficients, signature (10).
• Index to Elementary Cellular Automata
• Index entries for sequences related to cellular automata
• Index entries for sequences related to Benford's law

Crossrefs

Cf. A000005, A000012, A000290, A000400, A007953, A008776, A159991, A178500, A242614.

Cf. A178501: this sequence with 0 prefixed.

Row 5 of A329332.

Programs

Haskell

a011557 = (10 ^)
a011557_list = iterate (* 10) 1
-- Reinhard Zumkeller, Jul 05 2013, Feb 05 2012

Maple

A011557:=n->10^n; seq(A011557(n), n=0..40); # Wesley Ivan Hurt, Jan 17 2014

Mathematica

Table[10^n,{n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 15 2011 *)
10^Range[0,20] (* Harvey P. Dale, Sep 17 2023 *)

Maxima

A011557(n):=10^n$ makelist(A011557(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

PARI

a(n)=10^n \\ Charles R Greathouse IV, Jun 15 2011

Python

print([10**n for n in range(19)]) # Michael S. Branicky, Jan 10 2021

Formula

a(n) = 10^n.

a(n) = 10*a(n-1).

G.f.: 1/(1-10*x).

E.g.f.: exp(10*x).

A000005(a(n)) = A000290(n+1). - Reinhard Zumkeller, Mar 04 2007

a(n) = 60^n/6^n = A159991(n)/A000400(n). - Reinhard Zumkeller, May 02 2009

a(n) = A178501(n+1); for n > 0: a(n) = A178500(n). - Reinhard Zumkeller, May 28 2010

Sum_{n>0} 1/a(n) = 1/9 = A000012. - Stefano Spezia, Apr 28 2024

Extensions

Links to "Powers of Ten" books and videos added by N. J. A. Sloane, Nov 07 2009