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A102670 Number of digits >= 2 in the decimal representations of all integers from 0 to n.

%I A102670 #23 Nov 16 2022 17:27:31
%S A102670 0,0,1,2,3,4,5,6,7,8,8,8,9,10,11,12,13,14,15,16,17,18,20,22,24,26,28,
%T A102670 30,32,34,35,36,38,40,42,44,46,48,50,52,53,54,56,58,60,62,64,66,68,70,
%U A102670 71,72,74,76,78,80,82,84,86,88,89,90,92,94,96,98,100,102,104,106,107,108
%N A102670 Number of digits >= 2 in the decimal representations of all integers from 0 to n.
%C A102670 The total number of digits >= 2 occurring in all the numbers 0, 1, 2, ..., n (in decimal representation). - _Hieronymus Fischer_, Jun 10 2012
%H A102670 Hieronymus Fischer, <a href="/A102670/b102670.txt">Table of n, a(n) for n = 0..10000</a>
%F A102670 From _Hieronymus Fischer_, Jun 10 2012: (Start)
%F A102670 a(n) = (1/2)*Sum_{j=1..m+1} (floor(n/10^j + 0.8)*(2n + 2 + ((3/5) - floor(n/10^j + 4/5))*10^j) - floor(n/10^j)*(2n + 2 - (1 + floor(n/10^j)) * 10^j)), where m = floor(log_10(n)).
%F A102670 a(n) = (n+1)* A102669(n) + (1/2)*Sum_{j=1..m+1} (((3/5)*floor(n/10^j + 4/5) + floor(n/10^j))*10^j - (floor(n/10^j + 4/5)^2 - floor(n/10^j)^2)*10^j), where m = floor(log_10(n)).
%F A102670 a(10^m - 1) = 8*m*10^(m-1).
%F A102670 (This is the total number of digits >= 2 occurring in all the numbers with <= m places.)
%F A102670 G.f.: g(x) = (1/(1-x)^2)*Sum_{j>=0} (x^(2*10^j) - x^(10*10^j))/(1 - x^10^(j+1)).
%F A102670 General formulas for the total number of digits >= d in the decimal representations of all integers from 0 to n.
%F A102670 a(n) = (1/2)*Sum_{j=1..m+1} (floor(n/10^j + (10-d)/10) *(2n + 2 + ((5-d)/5 - floor(n/10^j + (10-d)/10))*10^j) - floor(n/10^j)*(2n + 2 - (1 + floor(n/10^j)) * 10^j)), where m = floor(log_10(n)).
%F A102670 a(n) = (n+1)*F(n,d) + (1/2)*Sum_{j=1..m+1} ((((5-d)/5)*floor(n/10^j + (10-d)/10) + floor(n/10^j))*10^j - (floor(n/10^j + (10-d)/10)^2 - floor(n/10^j)^2)*10^j), where m = floor(log_10(n)) and F(n,d) = number of digits >= d in the decimal representation of n.
%F A102670 a(10^m - 1) = (10-d)*m*10^(m-1).
%F A102670 (This is the total number of digits >= d occurring in all the numbers with <= m places.)
%F A102670 G.f.: g(x) = (1/(1-x)^2)*Sum_{j>=0} (x^(d*10^j) - x^(10*10^j))/(1 - x^10^(j+1)). (End)
%p A102670 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]>=2 then ct:=ct+1 else ct:=ct fi od: ct: end: seq(add(p(i),i=0..n), n=0..77); # _Emeric Deutsch_, Feb 23 2005
%t A102670 Accumulate[Table[Count[IntegerDigits[n],_?(#>1&)],{n,0,80}]] (* _Harvey P. Dale_, Apr 17 2014 *)
%Y A102670 Partial sums of A102669.
%Y A102670 Cf. A027868, A054899, A055640, A055641, A102669-A102685, A117804, A122840, A122841, A160093, A160094, A196563, A196564.
%Y A102670 Cf. A000120, A000788, A023416, A059015 (for base 2).
%K A102670 nonn,base,easy
%O A102670 0,4
%A A102670 _N. J. A. Sloane_, Feb 03 2005
%E A102670 More terms from _Emeric Deutsch_, Feb 23 2005