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a(n) = 10^n - 9^n + 1^n.
4

%I #22 Sep 08 2022 08:45:40

%S 1,2,20,272,3440,40952,468560,5217032,56953280,612579512,6513215600,

%T 68618940392,717570463520,7458134171672,77123207545040,

%U 794108867905352,8146979811148160,83322818300333432,849905364703000880

%N a(n) = 10^n - 9^n + 1^n.

%C a(n) is the number of numbers from 1 to 10^n containing a 1 in their decimal expansion. - _Charles R Greathouse IV_, Mar 25 2009

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

%F G.f.: 1/(1-10*x)-1/(1-9*x)+1/(1-x).

%F E.g.f.: e^(10*x)-e^(9*x)+e^x.

%F a(n) = 19*a(n-1)-90*a(n-2)+72 with a(0)=1, a(1)=2. - _Vincenzo Librandi_, Jul 21 2010

%F a(n) = 20*a(n-1)-109*a(n-2)+90*a(n-3). - _Wesley Ivan Hurt_, Apr 21 2021

%t Table[10^n-9^n+1,{n,0,20}] (* or *) LinearRecurrence[{20,-109,90},{1,2,20},20] (* _Harvey P. Dale_, Nov 05 2017 *)

%o (PARI) a(n)=10^n-9^n+1^n \\ _Charles R Greathouse IV_, Oct 16 2015

%o (Magma) [10^n-9^n+1: n in [0..25]]; // _Wesley Ivan Hurt_, Apr 21 2021

%Y Equals A016189 + 1. See that entry for more information.

%Y Cf. A155654, A155655, A155656, A155657, A155658, A155659, A155660, A155661, A155662, A155663, A155664, A155665, A155666, A155667, A155668, A155669, A155670.

%K nonn,easy

%O 0,2

%A _Mohammad K. Azarian_, Feb 01 2009