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%I M4653 N1992 #111 Feb 05 2024 10:55:13
%S 1,9,81,729,6561,59049,531441,4782969,43046721,387420489,3486784401,
%T 31381059609,282429536481,2541865828329,22876792454961,
%U 205891132094649,1853020188851841,16677181699666569,150094635296999121,1350851717672992089,12157665459056928801
%N Powers of 9: a(n) = 9^n.
%C 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.
%C Except for 1, the largest n-th power with n digits. - _Amarnath Murthy_, Feb 09 2002
%C 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
%C 1/1 + 1/9 + 1/81 + ... = 9/8. - _Gary W. Adamson_, Aug 29 2008
%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 9-colored compositions of n such that no adjacent parts have the same color. - _Milan Janjic_, Nov 17 2011
%C 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
%C For n > 0, a(n) is also the number of n-digit zeroless numbers (A052382). - _Stefano Spezia_, Jul 07 2022
%C Erasing the last digit of the sum a(n) + a(n+1) brings back a(n). - _Eric Angelini_, Feb 05 2024
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D 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.
%H T. D. Noe, <a href="/A001019/b001019.txt">Table of n, a(n) for n = 0..100</a>.
%H P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/groups.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=274">Encyclopedia of Combinatorial Structures 274</a>
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H R. J. Mathar, <a href="/A102518/a102518.pdf">Counting Walks on Finite Graphs</a>, Nov 2020, Section 4.
%H Y. Puri and T. Ward, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL4/WARD/short.html">Arithmetic and growth of periodic orbits</a>, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (9).
%F a(n) = 9^n.
%F a(0) = 1, a(n) = 9*a(n - 1) for n > 0.
%F G.f.: 1/(1 - 9*x).
%F E.g.f.: exp(9*x).
%F A000005(a(n)) = A005408(n + 1). - _Reinhard Zumkeller_, Mar 04 2007
%F a(n) = 4*A211866(n)+5. - _Reinhard Zumkeller_, Feb 12 2013
%F 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
%p A001019:=n->9^n: seq(A001019(n), n=0..25); # _Wesley Ivan Hurt_, Sep 27 2016
%t Table[9^n, {n, 0, 40}] (* _Vladimir Joseph Stephan Orlovsky_, Feb 15 2011 *)
%t NestList[9#&,1,20] (* _Harvey P. Dale_, Jul 04 2014 *)
%o (Maxima) A001019(n):=9^n$
%o makelist(A001019(n),n,0,30); /* _Martin Ettl_, Nov 05 2012 */
%o (Haskell)
%o a001019 = (9 ^)
%o a001019_list = iterate (* 9) 1
%o -- _Reinhard Zumkeller_, Feb 12 2013
%o (PARI) a(n)=9^n \\ _Charles R Greathouse IV_, Sep 24 2015
%o (Magma) [9^n : n in [0..25]]; // _Wesley Ivan Hurt_, Sep 27 2016
%Y Cf. A016189, A067470.
%Y Cf. A052382.
%K easy,nonn
%O 0,2
%A _N. J. A. Sloane_
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