A095807 - OEIS

%I #26 Feb 28 2023 04:09:44

%S 1,10,181,2620,33571,402130,4619161,51572440,564151951,6077367550,

%T 64696307941,682266771460,7140400943131,74263608488170,

%U 768372476393521,7915352287541680,81238170587875111

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

%H Vincenzo Librandi, <a href="/A095807/b095807.txt">Table of n, a(n) for n = 1..200</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (20,-109,90).

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

%F G.f.: (1-19*x+99*x^2)/((1-x)*(1-10*x)*(1-9*x)). - _Vincenzo Librandi_, Aug 14 2013

%F 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

%F Limit_{n->oo} a(n+1)/a(n) = 10. - _Bernard Schott_, Feb 28 2023

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

%t 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 *)

%o (PARI) a(n) = 10^n + 9/8 - 9^(1+n)/8; \\ _Michel Marcus_, Aug 13 2013

%o (Magma) [10^n + 9/8 - 9^(1+n)/8: n in [1..20]]; // _Vincenzo Librandi_, Aug 14 2013

%Y Cf. A016189.

%K nonn,easy,base

%O 1,2

%A _Michael Taktikos_, Aug 25 2004