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A102672 Number of digits >= 3 in the decimal representations of all integers from 0 to n.

%I A102672 #23 Nov 16 2022 17:27:01
%S A102672 0,0,0,1,2,3,4,5,6,7,7,7,7,8,9,10,11,12,13,14,14,14,14,15,16,17,18,19,
%T A102672 20,21,22,23,24,26,28,30,32,34,36,38,39,40,41,43,45,47,49,51,53,55,56,
%U A102672 57,58,60,62,64,66,68,70,72,73,74,75,77,79,81,83,85,87,89,90,91,92,94
%N A102672 Number of digits >= 3 in the decimal representations of all integers from 0 to n.
%C A102672 The total number of digits >= 3 occurring in all the numbers 0, 1, 2, ... n (in decimal representation). - _Hieronymus Fischer_, Jun 10 2012
%H A102672 Hieronymus Fischer, <a href="/A102672/b102672.txt">Table of n, a(n) for n = 0..10000</a>
%F A102672 From _Hieronymus Fischer_, Jun 10 2012: (Start)
%F A102672 a(n) = (1/2)*Sum_{j=1..m+1} (floor(n/10^j + 7/10)*(2n + 2 + (2/5 - floor(n/10^j + 7/10))*10^j) - floor(n/10^j)*(2n + 2 - (1 + floor(n/10^j)) * 10^j)), where m = floor(log_10(n)).
%F A102672 a(n) = (n+1)*A102671(n) + (1/2)*Sum_{j=1..m+1} (((2/5)*floor(n/10^j + 7/10) + floor(n/10^j))*10^j - (floor(n/10^j + 7/10)^2 - floor(n/10^j)^2)*10^j), where m = floor(log_10(n)).
%F A102672 a(10^m - 1) = 7*m*10^(m-1).
%F A102672 (This is the total number of digits >= 3 occurring in all the numbers with <= m places.)
%F A102672 G.f.: g(x) = (1/(1-x)^2)*Sum_{j>=0} (x^(3*10^j) - x^(10*10^j))/(1 - x^10^(j+1)). (End)
%p A102672 p:=proc(n) local b,ct,j: b:=convert(n,base,10): ct:=0: for j from 1 to nops(b) do if b[j]>=3 then ct:=ct+1 else ct:=ct fi od: ct: end: seq(add(p(i),i=0..n), n=0..80); # _Emeric Deutsch_, Feb 23 2005
%t A102672 Accumulate[Table[Count[IntegerDigits[n],_?(#>2&)],{n,0,80}]] (* _Harvey P. Dale_, Nov 23 2014 *)
%Y A102672 Partial sums of A102671.
%Y A102672 Cf. A027868, A054899, A055640, A055641, A102669-A102685, A117804, A122840, A122841, A160093, A160094, A196563, A196564. Partial sums of A102671.
%Y A102672 Cf. A000120, A000788, A023416, A059015 (for base 2).
%K A102672 nonn,base,easy
%O A102672 0,5
%A A102672 _N. J. A. Sloane_, Feb 03 2005
%E A102672 More terms from _Emeric Deutsch_, Feb 23 2005