A328356 a(n) is the sum of all positive integers whose decimal expansion is up to n digits and does not contain the 0 digit.
0, 45, 4500, 409095, 36855450, 3317322645, 298562027400, 26870609370195, 2418355085455350, 217651959870221745, 19588676407933119300, 1762980876890499197295, 158668278921733593899250, 14280145102970321446216845, 1285213059267457612117075200
Offset: 0
Examples
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. For n=3, the sum of all integers from 1 to 999 except those containing a zero is equal to a(3) = 409095.
Links
- Colin Barker, Table of n, a(n) for n = 0..500
- Pierre-Alain Sallard, Integer sequences A328348 and A328350 to A328356
- Index entries for linear recurrences with constant coefficients, signature (100,-909,810).
Programs
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PARI
concat(0, Vec(45*x / ((1 - x)*(1 - 9*x)*(1 - 90*x)) + O(x^15))) \\ Colin Barker, Dec 10 2019
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Python
[(80*90**n-89*9**n+9)*5//712 for n in range(20)]
Formula
a(n) = (80*90^n - 89*9^n + 9) * 5 / 712.
a(n) = 91*a(n-1) - 90*a(n-2) + 45*9^(n-1) for n > 1.
G.f.: 45*x / ((1 - x)*(1 - 9*x)*(1 - 90*x)). - Colin Barker, Dec 10 2019