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

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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).

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.

Programs

Haskell

a016189 n = 10 ^ n - 9 ^ n
a016189_list = 0 : zipWith (+) (map (* 9) a016189_list) a011557_list
-- Reinhard Zumkeller, Apr 03 2015

Magma

[10^n - 9^n: n in [0..20]]; // Vincenzo Librandi, Apr 26 2011

Mathematica

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

PARI

a(n)=10^n-9^n \\ M. F. Hasler, May 04 2015

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