A229127 Number of n-digit numbers containing the digit '0'.

1, 9, 171, 2439, 30951, 368559, 4217031, 46953279, 512579511, 5513215599, 58618940391, 617570463519, 6458134171671, 67123207545039, 694108867905351, 7146979811148159, 73322818300333431, 749905364703000879, 7649148282327007911, 77842334540943071199

Offset

1,2

Comments

Other than the number 0 itself, numbers with leading zeros are not allowed, so the general formula is a(n)=9*10^(n-1)-9^n, which is simply the number of n-digit numbers that begin with a nonzero digit (9*10^(n-1)) minus the number of n-digit numbers consisting only of nonzero digits (9^n). (Because of the 1-digit number 0 itself, the general formula does not apply at n=1.)

Other than the number 1, and 9 which is a semiprime, the minimum number of possible prime factors with multiplicity of a(n) = 3, which holds for 171 = 3^2 * 19; 2439 = 3^2 * 271; 46953279 = 3^2 * 5217031; 617570463519 = 3^2 * 68618940391; 77842334540943071199 = 3^2 * 8649148282327007911. - Jonathan Vos Post, Sep 16 2013

Links

Table of n, a(n) for n=1..20.

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

Formula

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

For n > 2, a(n) = 9*(a(n-1) + 10^(n-2)).

G.f.: x*(1-10*x+90*x^2)/((1-9*x)*(1-10*x)). - R. J. Mathar, Sep 14 2013

a(n) = A217094(n) - A217094(n-1), for n > 1. - Hieronymus Fischer, Dec 27 2013

E.g.f.: (9*exp(10*x) - 10*exp(9*x) + 10*x + 1)/10. - Stefano Spezia, Nov 15 2023

Example

a(2) = 9, since there are 9 2-digit numbers that contain a '0'.

Mathematica

LinearRecurrence[{19, -90}, {1, 9, 171}, 20] (* Stefano Spezia, Nov 15 2023 *)

Crossrefs

Cf. A011540, A050720, A217094.

Sequence in context: A121477 A055407 A083155 * A358620 A358959 A132897

Adjacent sequences: A229124 A229125 A229126 * A229128 A229129 A229130

Keyword

nonn,base,easy

Author

Jon E. Schoenfield, Sep 14 2013

Extensions

Example added and g.f. corrected by Hieronymus Fischer, Dec 27 2013

Status

approved