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%I #59 Nov 22 2022 03:10:01
%S 1,10,190,2890,38890,488890,5888890,68888890,788888890,8888888890,
%T 98888888890,1088888888890,11888888888890,128888888888890,
%U 1388888888888890,14888888888888890,158888888888888890,1688888888888888890,17888888888888888890,188888888888888888890
%N Number of zeros in numbers 0 to 999..9 (n digits).
%C This sequence also gives the total count of digits of n below 10^n. In such counts it makes sense to omit 10^0 as we are interested in having ten digits under each power of 10. For each power of 10 the total number of digits 0-9 is always the total of zeros for the next power. For example, at 10^1 there is 1 of each numeral 0-9, total 10 digits. At 10^2, the number of zeros is 10, with 20 each for the other 9 numerals and so on. - _Enoch Haga_, May 13 2006
%C Also the position of 10^n in Champernowne's constant (A033307). See Sikora, p. 3. - _Robert G. Wilson v_, Jun 29 2014
%H Vincenzo Librandi, <a href="/A033714/b033714.txt">Table of n, a(n) for n = 1..100</a>
%H John K. Sikora, <a href="http://arxiv.org/abs/1210.1263">On the High Water Mark Convergents of Champernowne's Constant in Base Ten</a>, arXiv:1210.1263 [math.NT], 2012.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (21,-120,100).
%F a(n) = 10^(n-1)*n - (1/9)*10^n + 10/9. - _Robert Israel_, Jun 30 2014
%F G.f.: -x*(100*x^2-11*x+1) / ((x-1)*(10*x-1)^2). - _Colin Barker_, Jan 27 2015
%F From _Bernard Schott_, Nov 20 2022: (Start)
%F a(n) = A033713(n) + 1.
%F a(n+1) = a(n) + 9 * A053541(n). (End)
%t a[1] = 1; a[n_] := a[n] = 9*10^(n-2)*(n-1) + a[n-1]; Table[a[n], {n, 1, 17}] (* _Jean-François Alcover_, Jul 13 2012 *)
%t f[n_] := 1 + Sum[9 m*10^(m - 1), {m, n}]; Array[f, 18, 0] (* _Robert G. Wilson v_, Jun 29 2014 *)
%t LinearRecurrence[{21,-120,100},{1,10,190},20] (* _Harvey P. Dale_, Dec 03 2021 *)
%o (Magma) [(9*n*10^n-10*10^n+100)/90: n in [1..20]]: // _Vincenzo Librandi_, Jul 01 2014
%o (PARI) Vec(-x*(100*x^2-11*x+1)/((x-1)*(10*x-1)^2) + O(x^100)) \\ _Colin Barker_, Jan 27 2015
%Y Cf. A033713, A053541.
%Y Cf. A212704 (first differences).
%K nonn,base,nice,easy
%O 1,2
%A Olivier Gorin (gorin(AT)roazhon.inra.fr)
%E More terms from _Erich Friedman_