A102685 - OEIS

%I #24 Nov 16 2022 14:53:25

%S 0,1,2,3,4,5,6,7,8,9,10,12,14,16,18,20,22,24,26,28,29,31,33,35,37,39,

%T 41,43,45,47,48,50,52,54,56,58,60,62,64,66,67,69,71,73,75,77,79,81,83,

%U 85,86,88,90,92,94,96,98,100,102,104,105,107,109,111,113,115,117,119,121,123

%N Partial sums of A055640.

%C The total number of nonzero digits occurring in all the numbers 0, 1, 2, ... n (in decimal representation). - _Hieronymus Fischer_, Jun 10 2012

%H Hieronymus Fischer, <a href="/A102685/b102685.txt">Table of n, a(n) for n = 0..10000</a>

%F From _Hieronymus Fischer_, Jun 06 2012: (Start)

%F a(n) = (1/2)*Sum_{j=1..m+1} (floor((n/10^j)+0.9)*(2n + 2 + (0.8 - floor((n/10^j)+0.9))*10^j) - floor(n/10^j)*(2n + 2 - (floor(n/10^j)+1) * 10^j)), where m = floor(log_10(n)).

%F a(n) = (n+1)*A055640(n) + (1/2)*Sum_{j=1..m+1} ((8*floor((n/10^j)+0.9)/10 + floor(n/10^j))*10^j - (floor((n/10^j)+0.9)^2 - floor(n/10^j)^2)*10^j), where m = floor(log_10(n)).

%F a(10^m-1) = 9*m*10^(m-1). (This is the total number of nonzero digits occurring in all the numbers with <= m digits.)

%F G.f.: g(x) = (1/(1-x)^2) * Sum_{j>=0} (x^10^j - x^(10*10^j))/(1-x^10^(j+1)). (End)

%Y Cf. A027868, A054899, A055640, A055641, A102669-A102684, A117804, A122840, A122841, A160093, A160094, A196563, A196564.

%Y Cf. A000120, A000788, A023416, A059015 (for base 2).

%K nonn

%O 0,3

%A _N. J. A. Sloane_, Feb 03 2005