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A113119 Total number of digits in all n-digit nonnegative integers.

%I A113119 #33 Dec 24 2022 19:04:24
%S A113119 10,180,2700,36000,450000,5400000,63000000,720000000,8100000000,
%T A113119 90000000000,990000000000,10800000000000,117000000000000,
%U A113119 1260000000000000,13500000000000000,144000000000000000,1530000000000000000,16200000000000000000,171000000000000000000
%N A113119 Total number of digits in all n-digit nonnegative integers.
%H A113119 Colin Barker, <a href="/A113119/b113119.txt">Table of n, a(n) for n = 1..950</a>
%H A113119 The Math Forum, <a href="https://web.archive.org/web/20021114194302/http://mathforum.org/library/drmath/view/59179.html">Ask Dr. Math: Really Counting to One Billion</a>, 2001.
%H A113119 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (20,-100).
%F A113119 For n > 1, a(n) = 9*n*10^(n-1).
%F A113119 From _Colin Barker_, Aug 05 2016: (Start)
%F A113119 a(n) = 20*a(n-1) - 100*a(n-2) for n > 3.
%F A113119 G.f.: 10*x*(1 - 2*x + 10*x^2) / (1-10*x)^2.
%F A113119 (End)
%F A113119 From _Bernard Schott_, Nov 14 2022: (Start)
%F A113119 a(n) = A212704(n) for n > 1.
%F A113119 a(n) = 9 * A053541(n) for n > 1.
%F A113119 a(n) = 9 * A081045(n-1) + A212704(n-1), for n > 1 (means a(n) = number of nonzero digits + number of zero digits). (End)
%F A113119 E.g.f.: x*(1 + 9*exp(10*x)). - _Stefano Spezia_, Dec 24 2022
%e A113119 a(1)=10 because there are ten one-digit numbers (including the 0).
%e A113119 a(2)=180 because there are 100-10=90 two-digit numbers, for a total of 90*2=180 digits.
%t A113119 LinearRecurrence[{20,-100},{10,180,2700},20] (* _Harvey P. Dale_, Dec 09 2021 *)
%o A113119 (PARI) Vec(10*x*(1-2*x+10*x^2)/(1-10*x)^2 + O(x^20)) \\ _Colin Barker_, Aug 05 2016
%o A113119 (Python)
%o A113119 def a(n): return 10 if n == 1 else 9*n*10**(n-1)
%o A113119 print([a(n) for n in range(1, 21)]) # _Michael S. Branicky_, Nov 14 2022
%Y A113119 Cf. A053541, A081045.
%Y A113119 Essentially the same as A212704.
%K A113119 base,easy,nonn
%O A113119 1,1
%A A113119 _Alexandre Wajnberg_, Jan 03 2006
%E A113119 More terms from _Joshua Zucker_, May 08 2006
%E A113119 a(17) corrected by _Colin Barker_, Aug 05 2016