A072290 Number of digits in the decimal expansion of the Champernowne constant that must be scanned to encounter all n-digit strings.

1, 11, 192, 2893, 38894, 488895, 5888896, 68888897, 788888898, 8888888899, 98888888900, 1088888888901, 11888888888902, 128888888888903, 1388888888888904, 14888888888888905, 158888888888888906, 1688888888888888907, 17888888888888888908, 188888888888888888909

Offset: 0

Author

Lekraj Beedassy, Jul 11 2002

Keywords

• nonn
• base
• easy

Comments

"Decimal expansion of the Champernowne constant" excludes the initial 0 to the left of the decimal point.

In writing out all numbers 1 through 10^n inclusive, exactly a(n) digits are used, of which a(n-1) are 0's and there are n*10^(n-1) of each of the other digits, with still an extra one for 1's.

References

• J. D. E. Konhauser et al. "Digit Counting." Problem 134 in Which Way Did The Bicycle Go? Dolciani Math. Exp. No. 18. Washington, DC: Math. Assoc. Amer., pp. 40 and 173-174, 1996.

Links

• Eric W. Weisstein, Table of n, a(n) for n = 0..1000 [replacing an earlier file from Vincenzo Librandi]
• Eric Weisstein's World of Mathematics, Champernowne Constant Digits
• Eric Weisstein's World of Mathematics, Constant Digit Scanning
• Index entries for linear recurrences with constant coefficients, signature (22,-141,220,-100).

Crossrefs

Cf. A078427.

Programs

Magma

[(10^(n-1)*n+n-10^n/9+1/9): n in [1..30]]; // Vincenzo Librandi, Jun 06 2011

Maple

A072290:=n->10/9 - 10^n/9 + n + n*10^n: seq(A072290(n), n=0..30); # Wesley Ivan Hurt, Jul 06 2014

Mathematica

f[n_] := 10/9 - 10^n/9 + n + n*10^n; Array[f, 20, 0] (* Robert G. Wilson v, Jul 06 2014 *)

PARI

for(n=1,23,print1(10^(n-1)*n+n-10^n/9+1/9" "));

PARI

Vec((91*x^2-11*x+1)/((x-1)^2*(10*x-1)^2) + O(x^100)) \\ Colin Barker, May 22 2014

Formula

a(n) = 10/9 - 10^n/9 + n + n*10^n.

a(n+1) = a(n) + 9*(n+1)*10^n + 1.

a(n+1) = n + A053541(n) - A002275(n) = n + A033713(n). - Lekraj Beedassy, Sep 16 2006

a(n) = 22*a(n-1) - 141*a(n-2) + 220*a(n-3) - 100*a(n-4). - Colin Barker, May 22 2014

G.f.: (91*x^2-11*x+1) / ((x-1)^2*(10*x-1)^2). - Colin Barker, May 22 2014

Extensions

More terms from Jason Earls, Dec 18 2002

Description rewritten by Eric W. Weisstein, Sep 14 2013

More terms from Colin Barker, May 22 2014