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

0,2

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).

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

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 *)

Prog

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

Crossrefs

Cf. A078427.

Sequence in context: A333759 A068649 A158509 * A258295 A213691 A112127

Adjacent sequences: A072287 A072288 A072289 * A072291 A072292 A072293

Keyword

nonn,base,easy

Author

Lekraj Beedassy, Jul 11 2002

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

Status

approved