A095807 Number of integers from 0 to 10^n - 1 whose decimal digits include at least one 0.

1, 10, 181, 2620, 33571, 402130, 4619161, 51572440, 564151951, 6077367550, 64696307941, 682266771460, 7140400943131, 74263608488170, 768372476393521, 7915352287541680, 81238170587875111

Offset

1,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Index entries for linear recurrences with constant coefficients, signature (20,-109,90).

Formula

a(n) = 10^n + 9/8 - 9^(1+n)/8.

G.f.: (1-19*x+99*x^2)/((1-x)*(1-10*x)*(1-9*x)). - Vincenzo Librandi, Aug 14 2013

a(n) = 20*a(n-1) - 109*a(n-2) + 90*a(n-3); a(0)=1, a(1)=10, a(2)=181. - Harvey P. Dale, Jun 20 2015

Limit_{n->oo} a(n+1)/a(n) = 10. - Bernard Schott, Feb 28 2023

Example

a(3)=181 because among the integers from 0 to 999 there are 181 numbers which contain at least 1 zero.

Mathematica

LinearRecurrence[{20, -109, 90}, {1, 10, 181}, 20] (* or *) Rest[ CoefficientList[ Series[(1-19x+99x^2)/((1-x)(1-10x)(1-9x)), {x, 0, 20}], x]] (* Harvey P. Dale, Jun 20 2015 *)

Prog

(PARI) a(n) = 10^n + 9/8 - 9^(1+n)/8; \\ Michel Marcus, Aug 13 2013
(Magma) [10^n + 9/8 - 9^(1+n)/8: n in [1..20]]; // Vincenzo Librandi, Aug 14 2013

Crossrefs

Cf. A016189.

Sequence in context: A030048 A318796 A054918 * A064092 A171513 A240405

Adjacent sequences: A095804 A095805 A095806 * A095808 A095809 A095810

Keyword

nonn,easy,base

Author

Michael Taktikos, Aug 25 2004

Status

approved