A033713 - OEIS

%I #28 Sep 13 2023 11:05:14

%S 0,9,189,2889,38889,488889,5888889,68888889,788888889,8888888889,

%T 98888888889,1088888888889,11888888888889,128888888888889,

%U 1388888888888889,14888888888888889,158888888888888889,1688888888888888889,17888888888888888889,188888888888888888889,1988888888888888888889

%N Number of zeros in numbers 1 to 999..9 (n digits).

%C Also the first n places of 1, ..., n-digit numbers in the almost-natural numbers (A007376). - _Erich Friedman_.

%C a(n+1) is also the total number of digits in numbers from 1 through 999..9 (n digits). - _Jianing Song_, Apr 17 2022

%D M. Kraitchik, Mathematical Recreations. Dover, NY, 2nd ed., 1953, p. 49.

%H F. Calogero, <a href="https://doi.org/10.1007/BF02984865">Cool irrational numbers and their rather cool rational approximations, Math. Intell. 25 (4), 72-76 (2003).

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (21,-120,100).

%F From _Stephen G Penrice_, Oct 01 2000: (Start)

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

%F G.f.: (9*x^2)/((1-x)(1-10x)^2). (End)

%F a(n) = Sum_{i=1..n} 9*i*10^(i-1).

%F a(1)=0, a(2)=9, a(3)=189, a(n)=21*a(n-1)-120*a(n-2)+100*a(n-3). - _Harvey P. Dale_, Jan 24 2012

%F a(n+1) = A058183(10^n-1) for n >= 1. - _Jianing Song_, Apr 17 2022

%F E.g.f.: exp(x)*(1 + exp(9*x)*(9*x - 1))/9. - _Stefano Spezia_, Sep 13 2023

%t Table[ Sum[9i*10^(i - 1), {i, 1, n}], {n, 0, 16}]

%t LinearRecurrence[{21,-120,100},{0,9,189},30] (* _Harvey P. Dale_, Jan 24 2012 *)

%o (PARI) a(n)=((n-1)*(10^n)-n*10^(n-1)+1)/9 \\ _Charles R Greathouse IV_, Feb 19 2017

%Y Cf. A033714, A058183.

%K nonn,base,nice,easy

%O 1,2

%A Olivier Gorin (gorin(AT)roazhon.inra.fr)

%E More terms from _Erich Friedman_.

%E a(18)-a(21) from _Stefano Spezia_, Sep 13 2023