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A102684
Number of times the digit 9 appears in the decimal representations of all integers from 0 to n.
4
0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20
OFFSET
0,20
COMMENTS
This is the total number of digits = 9 occurring in all the numbers 0, 1, 2, ... n (in decimal representation). - Hieronymus Fischer, Jun 10 2012
LINKS
FORMULA
From Hieronymus Fischer, Jun 10 2012: (Start)
a(n) = (1/2)*Sum_{j=1..m+1} (floor(n/10^j + 1/10)*(2n + 2 - (4/5 + floor(n/10^j + 1/10))*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)*A102683(n) + (1/2)*Sum_{j=1..m+1} ((-4/5*floor(n/10^j + 1/10) + floor(n/10^j))*10^j - (floor(n/10^j + 1/10)^2 - floor(n/10^j)^2)*10^j), where m = floor(log_10(n)).
a(10^m-1) = m*10^(m-1).
(this is total number of digits = 9 occurring in all the numbers with <= m places).
G.f.: g(x) = (1/(1-x)^2)*Sum_{j>=0} (x^(9*10^j) - x^(10*10^j))/(1-x^10^(j+1)). (End)
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]>=9 then ct:=ct+1 else ct:=ct fi od: ct: end: seq(add(p(i), i=0..n), n=0..105); # Emeric Deutsch, Feb 23 2005
MATHEMATICA
Accumulate[DigitCount[Range[0, 100], 10, 9]] (* Harvey P. Dale, Mar 30 2018 *)
PROG
(PARI) a(n) = sum(k=0, n, #select(x->(x==9), digits(k))); \\ Michel Marcus, Oct 03 2023
CROSSREFS
KEYWORD
nonn,base,easy
AUTHOR
N. J. A. Sloane, Feb 03 2005
EXTENSIONS
More terms from Emeric Deutsch, Feb 23 2005
Definition revised by N. J. A. Sloane, Mar 30 2018
STATUS
approved