A016189 a(n) = 10^n - 9^n.

0, 1, 19, 271, 3439, 40951, 468559, 5217031, 56953279, 612579511, 6513215599, 68618940391, 717570463519, 7458134171671, 77123207545039, 794108867905351, 8146979811148159, 83322818300333431, 849905364703000879, 8649148282327007911, 87842334540943071199, 890581010868487640791

Offset

0,3

Comments

Almost all numbers contain any given sequence of digits (in any base) [Theorem 143 of Hardy and Wright]. a(7) = 5217031, more than 52% of the numbers < 10^7 contain any given nonzero decimal digit. - Frank Ellermann, May 30 2001

a(n) gives the number of integers from 0 to 10^n-1 which contain (at least) any one given decimal digit except 0. - Michael Taktikos, Aug 24 2004

These are the numerators of a(n)=(integral{x=0 to 0.2} (1-0.5*x)^n dx). E.g., a(3)=3439/20000. The denominators are b(n)=5*(n+1)*10^n. E.g., b(3)=20000. - Al Hakanson (hawkuu(AT)excite.com), Feb 22 2004

Binomial transforms of sequences defined by a(n)=(C+1)^n-C^n are the sequences (C+2)^n-(C+1)^n. The binomial transform of this here is in A016195, for example. - R. J. Mathar, Nov 27 2008

First differences are given in A088924. - M. F. Hasler, May 04 2015

References

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 143

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..130

Alexander Bogomolny, Almost every integer has a digit 3 in it

John Elias, Illustration of Initial Terms

James Grime, 3 is everywhere, Numberphile video

Index entries for linear recurrences with constant coefficients, signature (19, -90).

Formula

G.f.: x/((1-9x)(1-10x)).

a(0) = 0, a(1) = 1, then a(n+1) = 9*a(n) + 10^n.

a(n) = 19*a(n-1) - 90*a(n-2), n > 1; a(0)=0, a(1)=1. - Philippe Deléham, Jan 01 2009

E.g.f.: e^(10*x) - e^(9*x). - Mohammad K. Azarian, Jan 14 2009

Mathematica

f[n_]:=10^n-9^n; f[Range[0, 40]] (* Vladimir Joseph Stephan Orlovsky, Feb 14 2011 *)

Prog

(Magma) [10^n - 9^n: n in [0..20]]; // Vincenzo Librandi, Apr 26 2011
(Haskell)
a016189 n = 10 ^ n - 9 ^ n
a016189_list = 0 : zipWith (+) (map (* 9) a016189_list) a011557_list
-- Reinhard Zumkeller, Apr 03 2015
(PARI) a(n)=10^n-9^n \\ M. F. Hasler, May 04 2015

Crossrefs

Base 2: A000225, 3: A001047, 4: A005061, 5: A005060, 6: A005062, base 7: A016169, 8: A016177, 9: A016185 11: A016195 12: A016197.

Equals A155671 - 1.

Cf. A011557, A011533.

Sequence in context: A142899 A083004 A139739 * A125476 A016248 A322539

Adjacent sequences: A016186 A016187 A016188 * A016190 A016191 A016192

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved