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A102684 Number of times the digit 9 appears in the decimal representations of all integers from 0 to n. 3
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 (list; graph; refs; listen; history; text; internal format)
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

Hieronymus Fischer, Table of n, a(n) for n = 0..10000

FORMULA

Contribution 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 *)

CROSSREFS

Partial sums of A102683.

Cf. A027868, A054899, A055640, A055641, A102669-A102685, A117804, A122840, A122841, A160093, A160094, A196563, A196564.

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

Sequence in context: A179051 A054899 A061217 * A156821 A025856 A103378

Adjacent sequences:  A102681 A102682 A102683 * A102685 A102686 A102687

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

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Last modified November 17 20:47 EST 2018. Contains 317278 sequences. (Running on oeis4.)