A102674 Number of digits >= 4 in the decimal representations of all integers from 0 to n.
0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 6, 6, 6, 6, 7, 8, 9, 10, 11, 12, 12, 12, 12, 12, 13, 14, 15, 16, 17, 18, 18, 18, 18, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 30, 32, 34, 36, 38, 40, 41, 42, 43, 44, 46, 48, 50, 52, 54, 56, 57, 58, 59, 60, 62, 64, 66, 68, 70, 72, 73, 74, 75, 76, 78
Offset: 0
Links
- Hieronymus Fischer, Table of n, a(n) for n = 0..10000
Crossrefs
Programs
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Maple
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]>=4 then ct:=ct+1 else ct:=ct fi od: ct: end:seq(add(p(i),i=0..n),n=0..90); # Emeric Deutsch, Feb 22 2005
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Mathematica
Accumulate[Table[Total[Drop[Most[DigitCount[n]],3]],{n,0,80}]] (* Harvey P. Dale, Nov 27 2015 *)
Formula
From Hieronymus Fischer, Jun 10 2012: (Start)
a(n) = (1/2)*Sum_{j=1..m+1} (floor(n/10^j + 3/5)*(2n + 2 + (1/5 - floor(n/10^j + 3/5))*10^j) - floor(n/10^j)*(2n + 2 - (1 + floor(n/10^j)) * 10^j)), where m = floor(log_10(n)).
a(n) = (n+1)*A102673(n) + (1/2)*Sum_{j=1..m+1} (((1/5)*floor(n/10^j + 3/5) + floor(n/10^j))*10^j - (floor(n/10^j + 3/5)^2 - floor(n/10^j)^2)*10^j), where m = floor(log_10(n)).
a(10^m - 1) = 6*m*10^(m-1).
(This is the total number of digits >= 4 occurring in all the numbers with <= m places.)
G.f.: g(x) = (1/(1-x)^2)*Sum_{j>=0} (x^(4*10^j) - x^(10*10^j))/(1 - x^10^(j+1)). (End)
Extensions
More terms from Emeric Deutsch, Feb 22 2005
Comments