

A001019


Powers of 9: a(n) = 9^n.
(Formerly M4653 N1992)


105



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

0,2


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 nth power with n digits.  Amarnath Murthy, Feb 09 2002
The 2002 comment by Amarnath Murthy should say more precisely "nth 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
The compositions of n in which each natural number is colored by one of p different colors are called pcolored compositions of n. For n>=1, a(n) equals the number of 9colored 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 nth 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


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



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).
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


MAPLE



MATHEMATICA



PROG

(Haskell)
a001019 = (9 ^)
a001019_list = iterate (* 9) 1


CROSSREFS



KEYWORD

easy,nonn


AUTHOR



STATUS

approved



