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A268839 a(n) = Sum_{j=1..10^n-1} 2^f(j) where f(j) is the number of zero digits in the decimal representation of j.

%I A268839 #30 Sep 13 2023 11:58:33
%S A268839 9,108,1197,13176,144945,1594404,17538453,192922992,2122152921,
%T A268839 23343682140,256780503549,2824585539048,31070440929537,
%U A268839 341774850224916,3759523352474085,41354756877214944,454902325649364393,5003925582143008332,55043181403573091661
%N A268839 a(n) = Sum_{j=1..10^n-1} 2^f(j) where f(j) is the number of zero digits in the decimal representation of j.
%C A268839 We calculate the number of integers between 1 and 10^n - 1 having k zeros in their decimal representation. To form a such number consisting of m digits (k < m), place k zeros in m-1 possible positions, then we must choose m-k digits different from zero. Thus, the number of integers between 1 and 10^n - 1 having k zeros in their decimal representation is: Sum_{m=k+1..n} binomial(m-1, k)*9^(m-k).
%C A268839 Hence the sum: Sum_{m=1..n} Sum_{k=0..m-1} binomial(m-1,k)*9^(m-k)*2^k = Sum_{m=1..n} 9^m*(11/9)*(m-1) = (9/10)*(11^n - 1).
%H A268839 Colin Barker, <a href="/A268839/b268839.txt">Table of n, a(n) for n = 1..950</a>
%H A268839 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (12,-11).
%F A268839 a(n) = (9/10)*(11^n-1) = 9*A016123(n-1).
%F A268839 From _Vincenzo Librandi_, Feb 15 2016: (Start)
%F A268839 G.f.: (9*x)/((1-11*x)*(1-x)).
%F A268839 a(n) = 11*a(n-1) + 9. (End)
%F A268839 E.g.f.: 9*exp(x)*(exp(10*x) - 1)/10. - _Stefano Spezia_, Sep 13 2023
%e A268839 a(1) = 9 because 2^f(1) + 2^f(2) + ... + 2^f(9) = 2^0 + 2^0 + ... + 2^0 = 9;
%e A268839 a(2) = 108 because 2^f(1) + 2^f(2) + ... + 2^f(99) = 9*10 + 2*9 = 108, where f(10) = f(20) = ... = f(90) = 1 and f(i) = 0 otherwise.
%p A268839 for n from 1 to 100 do: x:=(9/10)*(11^n-1):printf(‘%d, ‘,x):od:
%t A268839 Table[Table[(9/10) (11^n - 1), {n, 1, 20}]] (* _Bruno Berselli_, Feb 15 2016 *)
%t A268839 CoefficientList[Series[9/((1 - 11 x) (1 - x)), {x, 0, 33}], x] (* _Vincenzo Librandi_, Feb 15 2016 *)
%o A268839 (Magma) [(9/10)*(11^n-1): n in [1..20]]; // _Vincenzo Librandi_, Feb 15 2016
%o A268839 (PARI) Vec(9*x/((1-11*x)*(1-x)) + O(x^30)) \\ _Colin Barker_, Feb 22 2016
%Y A268839 Cf. A016123, A033713, A033714, A055641, A119291.
%K A268839 nonn,base,easy
%O A268839 1,1
%A A268839 _Michel Lagneau_, Feb 14 2016
%E A268839 Name edited by _Jon E. Schoenfield_, Sep 13 2017