A001019 Powers of 9: a(n) = 9^n.

1, 9, 81, 729, 6561, 59049, 531441, 4782969, 43046721, 387420489, 3486784401, 31381059609, 282429536481, 2541865828329, 22876792454961, 205891132094649, 1853020188851841, 16677181699666569, 150094635296999121, 1350851717672992089, 12157665459056928801

Offset: 0

Author

N. J. A. Sloane

Keywords

• easy
• nonn

Comments

Same as Pisot sequences E(1, 9), L(1, 9), P(1, 9), T(1, 9). Essentially same as Pisot sequences E(9, 81), L(9, 81), P(9, 81), T(9, 81). See A008776 for definitions of Pisot sequences.

Except for 1, the largest n-th power with n digits. - Amarnath Murthy, Feb 09 2002

The 2002 comment by Amarnath Murthy should say more precisely "n-th power with *at most* n digits": a(22) has only 21 digits etc., a(44) has only 42 digits etc. - Hagen von Eitzen, May 17 2009

1/1 + 1/9 + 1/81 + ... = 9/8. - Gary W. Adamson, Aug 29 2008

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

To be still more precise than Murthy and von Eitzen: the subsequence of the largest n-th power with n digits is a finite sequence, bounded by 9 and 109418989131512359209. It is guaranteed that 10^n has n + 1 digits in base 10, and clearly 9^n < 10^n. With a(22), the number n - log_10 a(n) crosses the 1.0 threshold, and obviously the gulf widens further after that, meaning that for n > 21, m^n can have fewer than n digits or more than n digits but not exactly n digits. - Alonso del Arte, Dec 12 2012

For n > 0, a(n) is also the number of n-digit zeroless numbers (A052382). - Stefano Spezia, Jul 07 2022

Erasing the last digit of the sum a(n) + a(n+1) brings back a(n). - Eric Angelini, Feb 05 2024

References

• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
• David Wells, The Penguin Dictionary of Curious and Interesting Integers. London: Penguin Books (1997): p. 196, entry for 109,418,989,131,512,359,209.

Links

• T. D. Noe, Table of n, a(n) for n = 0..100.
• P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 274
• Tanya Khovanova, Recursive Sequences
• R. J. Mathar, Counting Walks on Finite Graphs, Nov 2020, Section 4.
• Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
• Index entries for linear recurrences with constant coefficients, signature (9).

Crossrefs

Cf. A016189, A067470.

Cf. A052382.

Programs

Haskell

a001019 = (9 ^)
a001019_list = iterate (* 9) 1
-- Reinhard Zumkeller, Feb 12 2013

Magma

[9^n : n in [0..25]]; // Wesley Ivan Hurt, Sep 27 2016

Maple

A001019:=n->9^n: seq(A001019(n), n=0..25); # Wesley Ivan Hurt, Sep 27 2016

Mathematica

Table[9^n, {n, 0, 40}] (* Vladimir Joseph Stephan Orlovsky, Feb 15 2011 *)
NestList[9#&,1,20] (* Harvey P. Dale, Jul 04 2014 *)

Maxima

A001019(n):=9^n$
makelist(A001019(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

PARI

a(n)=9^n \\ Charles R Greathouse IV, Sep 24 2015

Formula

a(n) = 9^n.

a(0) = 1, a(n) = 9*a(n - 1) for n > 0.

G.f.: 1/(1 - 9*x).

E.g.f.: exp(9*x).

A000005(a(n)) = A005408(n + 1). - Reinhard Zumkeller, Mar 04 2007

a(n) = 4*A211866(n)+5. - Reinhard Zumkeller, Feb 12 2013

a(n) = det(|v(i+2,j)|, 1 <= i,j <= n), where v(n,k) are central factorial numbers of the first kind with odd indices. - Mircea Merca, Apr 04 2013