A052268 First differences of 10^n (A011557).

9, 90, 900, 9000, 90000, 900000, 9000000, 90000000, 900000000, 9000000000, 90000000000, 900000000000, 9000000000000, 90000000000000, 900000000000000, 9000000000000000, 90000000000000000, 900000000000000000, 9000000000000000000, 90000000000000000000

Offset: 1

Author

Barry E. Williams, Feb 03 2000

Keywords

• easy
• nonn

Comments

a(n) is equal to the number of functions f:{1,2...,n}->{1,2,...,10} such that for a fixed x in {1,2,...,n} and a fixed y in {1,2,...,10} we have f(x)<>y. - Aleksandar M. Janjic and Milan Janjic, Mar 27 2007

a(n) is the number of n-digit positive integers. - Geoffrey Critzer, Apr 23 2009

References

• A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
• Miklos Bona, Introduction to Enumerative Combinatorics, McGraw-Hill, 2007, p. 8.

Links

• Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
• Index entries for linear recurrences with constant coefficients, signature (10).

Crossrefs

Cf. A011557.

Programs

Mathematica

q = 10; Join[{a = 1}, Table[If[n == 0, a = q * a - 1, a = q * a], {n, 0, 25}]] (* Vladimir Joseph Stephan Orlovsky, Jul 11 2011 *)
Differences[10^Range[0, 19]] (* Alonso del Arte, Feb 23 2015 *)

PARI

a(n)=9*10^(n-1) \\ Charles R Greathouse IV, Sep 24 2015

Formula

a(n) = 9*10^(n-1).

From Stefano Spezia, Jun 03 2021: (Start)

O.g.f.: 9*x/(1 - 10*x).

E.g.f.: 9*(exp(10*x) - 1)/10.

a(n) = 10*a(n-1) for n > 1. (End)

Extensions

Deleted erroneous term a(0)=1. - N. J. A. Sloane, Apr 02 2015