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%I #19 Dec 12 2019 09:33:10
%S 0,45,4500,409095,36855450,3317322645,298562027400,26870609370195,
%T 2418355085455350,217651959870221745,19588676407933119300,
%U 1762980876890499197295,158668278921733593899250,14280145102970321446216845,1285213059267457612117075200
%N a(n) is the sum of all positive integers whose decimal expansion is up to n digits and does not contain the 0 digit.
%H Colin Barker, <a href="/A328356/b328356.txt">Table of n, a(n) for n = 0..500</a>
%H Pierre-Alain Sallard, <a href="/A328356/a328356.pdf">Integer sequences A328348 and A328350 to A328356</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (100,-909,810).
%F a(n) = (80*90^n - 89*9^n + 9) * 5 / 712.
%F a(n) = 91*a(n-1) - 90*a(n-2) + 45*9^(n-1) for n > 1.
%F G.f.: 45*x / ((1 - x)*(1 - 9*x)*(1 - 90*x)). - _Colin Barker_, Dec 10 2019
%e For n=2, the sum of all integers from 1 to 99 except those containing a zero (i.e., except multiples of 10: 10, 20, ..., 90) is equal to a(2) = 4500.
%e For n=3, the sum of all integers from 1 to 999 except those containing a zero is equal to a(3) = 409095.
%o (Python) [(80*90**n-89*9**n+9)*5//712 for n in range(20)]
%o (PARI) concat(0, Vec(45*x / ((1 - x)*(1 - 9*x)*(1 - 90*x)) + O(x^15))) \\ _Colin Barker_, Dec 10 2019
%Y Cf. A328348, A328350, A328351, A328352, A328353, A328354, A328355.
%K nonn,base,easy
%O 0,2
%A _Pierre-Alain Sallard_, Dec 10 2019
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