A072290 -id:A072290 - cp's OEIS frontend

A078427 Sum of all the decimal digits of numbers from 1 to 10^n.

46, 901, 13501, 180001, 2250001, 27000001, 315000001, 3600000001, 40500000001, 450000000001, 4950000000001, 54000000000001, 585000000000001, 6300000000000001, 67500000000000001, 720000000000000001, 7650000000000000001, 81000000000000000001

Offset: 1

Shyam Sunder Gupta, Dec 29 2002

			a(2)=901 because sum of all the digits of numbers from 1 to 10^2 is 901.

E. J. Barbeau et al., Five Hundred Mathematical Challenges, Problem 284. pp. 25; 142-143, MAA Washington DC, 1995.

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Index entries for linear recurrences with constant coefficients, signature (21,-120,100).

Cf. A053541, A072290.

[(45*n)*10^(n-1)+1: n in [1..30]]; // Vincenzo Librandi, Jun 06 2011

LinearRecurrence[{21,-120,100},{46,901,13501},20] (* Harvey P. Dale, Nov 24 2016 *)

Vec(-x*(100*x^2-65*x+46)/((x-1)*(10*x-1)^2) + O(x^100)) \\ Colin Barker, May 23 2014

a(n) = (45*n)*10^(n-1)+1.
a(n) = 45*A053541(n)+1. - Lekraj Beedassy, Sep 16 2006
a(n) = 21*a(n-1)-120*a(n-2)+100*a(n-3). - Colin Barker, May 23 2014
G.f.: -x*(100*x^2-65*x+46) / ((x-1)*(10*x-1)^2). - Colin Barker, May 23 2014

Edited by Charles R Greathouse IV, Aug 02 2010

More terms from Colin Barker, May 23 2014

A229186 Beginning position of n in the decimal expansion of the Champernowne constant.

Original entry on oeis.org

11, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 1, 16, 18, 20, 22, 24, 26, 28, 30, 15, 34, 2, 38, 40, 42, 44, 46, 48, 50, 17, 37, 56, 3, 60, 62, 64, 66, 68, 70, 19, 39, 59, 78, 4, 82, 84, 86, 88, 90, 21, 41, 61, 81, 100, 5, 104, 106, 108, 110, 23

Offset: 0

Eric W. Weisstein, Sep 15 2013

Ignoring the initial 0 to the left of the decimal point.

Same as A031297 but including the a(0) term.

Eric W. Weisstein, Table of n, a(n) for n = 0..999
Eric Weisstein's World of Mathematics, Champernowne Constant Digits
Eric Weisstein's World of Mathematics, Constant Digit Scanning

Cf. A033307 (decimal expansion of the Champernowne constant).
Cf. A072290 (number of digits in the decimal expansion of the Champernowne constant that must be scanned to encounter all n-digit strings).
Cf. A031297 (same sequence but omitting the a(0) term).

from itertools import count, islice
def agen():
    k, chap = 1, ".1"
    for n in count(0):
        target = str(n)
        while chap.find(target) == -1: k += 1; chap += str(k)
        yield chap.find(target)
print(list(islice(agen(), 70))) # Michael S. Branicky, Oct 06 2022

A094797 Number of times 1 is used in writing out all numbers 1 through 10^n.

Original entry on oeis.org

1, 2, 21, 301, 4001, 50001, 600001, 7000001, 80000001, 900000001, 10000000001, 110000000001, 1200000000001, 13000000000001, 140000000000001, 1500000000000001, 16000000000000001, 170000000000000001, 1800000000000000001, 19000000000000000001

Offset: 0

Lekraj Beedassy, Jun 11 2004

Index entries for linear recurrences with constant coefficients, signature (21,-120,100).

Cf. A072290, A078427.

Cf. A011557, A094798.

Table[ n*10^(n - 1) + 1, {n, 0, 17}] (* Robert G. Wilson v, Jun 15 2004 *)
LinearRecurrence[{21,-120,100},{1,2,21},20] (* Harvey P. Dale, Sep 07 2022 *)

Vec(-(99*x^2-19*x+1)/((x-1)*(10*x-1)^2) + O(x^100)) \\ Colin Barker, May 23 2014

a(n) = n*10^(n-1) + 1.
a(n) = 21*a(n-1)-120*a(n-2)+100*a(n-3). - Colin Barker, May 23 2014
G.f.: -(99*x^2-19*x+1) / ((x-1)*(10*x-1)^2). - Colin Barker, May 23 2014
a(n) = A094798(A011557(n)). - Michel Marcus, Oct 03 2023

More terms from Robert G. Wilson v, Jun 15 2004

Further terms from Colin Barker, May 23 2014

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A078427 Sum of all the decimal digits of numbers from 1 to 10^n.

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A229186 Beginning position of n in the decimal expansion of the Champernowne constant.

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A094797 Number of times 1 is used in writing out all numbers 1 through 10^n.

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Mathematica

PARI

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