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A268839 a(n) = Sum_{j=1..10^n-1} 2^f(j) where f(j) is the number of zero digits in the decimal representation of j. 1
9, 108, 1197, 13176, 144945, 1594404, 17538453, 192922992, 2122152921, 23343682140, 256780503549, 2824585539048, 31070440929537, 341774850224916, 3759523352474085, 41354756877214944, 454902325649364393, 5003925582143008332, 55043181403573091661 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

We calculate the number of integers between 1 and 10^n - 1 having k zeros in their decimal representation. To form a such number consisting of m digits (k < m), place k zeros in m-1 possible positions, then we must choose m-k digits different from zero. Thus, the number of integers between 1 and 10^n - 1 having k zeros in their decimal representation is: Sum_{m=k+1..n} binomial(m-1, k)*9^(m-k).

Hence the sum: Sum_{m=1..n} Sum_{k=0..m-1} binomial(m-1,k)*9^(m-k)*2^k = Sum_{m=1..n} 9^m*(11/9)*(m-1) = (9/10)*(11^n - 1).

LINKS

Colin Barker, Table of n, a(n) for n = 1..950

Index entries for linear recurrences with constant coefficients, signature (12,-11).

FORMULA

a(n) = (9/10)*(11^n-1) = 9*A016123(n-1).

G.f.: (9*x)/((1-11*x)*(1-x)). - Vincenzo Librandi, Feb 15 2016

a(n) = 11*a(n-1) + 9. - Vincenzo Librandi, Feb 15 2016

EXAMPLE

a(1) = 9 because 2^f(1) + 2^f(2) + ... + 2^f(9) = 2^0 + 2^0 + ... + 2^0 = 9;

a(2) = 108 because 2^f(1) + 2^f(2) + ... + 2^f(99) = 9*10 + 2*9 = 108, where f(10) = f(20) = ... = f(90) = 1 and f(i) = 0 otherwise.

MAPLE

for n from 1 to 100 do: x:=(9/10)*(11^n-1):printf(‘%d, ‘, x):od:

MATHEMATICA

Table[Table[(9/10) (11^n - 1), {n, 1, 20}]] (* Bruno Berselli, Feb 15 2016 *)

CoefficientList[Series[9/((1 - 11 x) (1 - x)), {x, 0, 33}], x] (* Vincenzo Librandi, Feb 15 2016 *)

PROG

(MAGMA) [(9/10)*(11^n-1): n in [1..20]]; // Vincenzo Librandi, Feb 15 2016

(PARI) Vec(9*x/((1-11*x)*(1-x)) + O(x^30)) \\ Colin Barker, Feb 22 2016

CROSSREFS

Cf. A016123, A033713, A033714, A055641, A119291.

Sequence in context: A080505 A104224 A099676 * A214668 A234467 A288550

Adjacent sequences:  A268836 A268837 A268838 * A268840 A268841 A268842

KEYWORD

nonn,base,easy

AUTHOR

Michel Lagneau, Feb 14 2016

EXTENSIONS

Name edited by Jon E. Schoenfield, Sep 13 2017

STATUS

approved

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Last modified October 18 05:26 EDT 2019. Contains 328146 sequences. (Running on oeis4.)