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%I #56 Aug 08 2023 16:35:28
%S 1,20,300,4000,50000,600000,7000000,80000000,900000000,10000000000,
%T 110000000000,1200000000000,13000000000000,140000000000000,
%U 1500000000000000,16000000000000000,170000000000000000
%N a(n) = n*10^(n-1).
%C This sequence gives the number of 1's (or any other nonzero digit) required to write all integers from 0 up to 10^n-1. - Jason D. W. Taff (jtaff(AT)jburroughs.org), Dec 05 2004 (improved by _Bernard Schott_, Nov 17 2022)
%C The corresponding number of 0's required to write all these integers from 0 up to 10^n-1 is A033714(n). - _Bernard Schott_, Nov 17 2022
%D Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
%H Vincenzo Librandi, <a href="/A053541/b053541.txt">Table of n, a(n) for n = 1..100</a>
%H Frank Ellermann, <a href="/A001792/a001792.txt">Illustration of binomial transforms</a>.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (20,-100).
%F a(n) = 20*a(n-1) - 100*a(n-2), with a(0)=0, a(1)=1, a(2)=20.
%F From Jason D. W. Taff (jtaff(AT)jburroughs.org), Dec 05 2004: (Start)
%F a(n) = 10*a(n-1) + 10*(n-1).
%F a(n) = Sum_{k=1..n} k*binomial(n,k)*9^(n-k).
%F a(n) = A094798(10^n - 1). (End)
%F From _G. C. Greubel_, May 16 2019: (Start)
%F G.f.: x/(1-10*x)^2.
%F E.g.f.: x*exp(10*x). (End)
%F From _Amiram Eldar_, Oct 28 2020: (Start)
%F Sum_{n>=1} 1/a(n) = 10*log(10/9).
%F Sum_{n>=1} (-1)^(n+1)/a(n) = 10*log(11/10). (End)
%F a(n) = Sum_{k=1..n} A081045(k-1). - _Bernard Schott_, Nov 17 2022
%p seq(n*10^(n-1), n = 1 .. 40); # _Bernard Schott_, Nov 17 2022
%t f[n_]:=n*10^(n-1);f[Range[40]] (* _Vladimir Joseph Stephan Orlovsky_, Feb 09 2011*)
%t LinearRecurrence[{20,-100},{1,20},20] (* _Harvey P. Dale_, Aug 08 2023 *)
%o (Magma) [n*10^(n-1): n in [1..30]]; // _Vincenzo Librandi_, Jun 06 2011
%o (PARI) a(n)=n*10^(n-1) \\ _Charles R Greathouse IV_, Dec 05 2011
%o (Sage) [n*10^(n-1) for n in (1..20)] # _G. C. Greubel_, May 16 2019
%o (GAP) List([1..20], n-> n*10^(n-1)) # _G. C. Greubel_, May 16 2019
%Y Cf. A001787, A033714, A038303, A053464, A053469, A081045, A094798.
%K easy,nonn
%O 1,2
%A _Barry E. Williams_, Jan 15 2000
%E Offset changed from 0 to 1 by _Vincenzo Librandi_, Jun 06 2011
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