OFFSET
1,2
COMMENTS
This sequence also gives the total count of digits of n below 10^n. In such counts it makes sense to omit 10^0 as we are interested in having ten digits under each power of 10. For each power of 10 the total number of digits 0-9 is always the total of zeros for the next power. For example, at 10^1 there is 1 of each numeral 0-9, total 10 digits. At 10^2, the number of zeros is 10, with 20 each for the other 9 numerals and so on. - Enoch Haga, May 13 2006
Also the position of 10^n in Champernowne's constant (A033307). See Sikora, p. 3. - Robert G. Wilson v, Jun 29 2014
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..100
John K. Sikora, On the High Water Mark Convergents of Champernowne's Constant in Base Ten, arXiv:1210.1263 [math.NT], 2012.
Index entries for linear recurrences with constant coefficients, signature (21,-120,100).
FORMULA
a(n) = 10^(n-1)*n - (1/9)*10^n + 10/9. - Robert Israel, Jun 30 2014
G.f.: -x*(100*x^2-11*x+1) / ((x-1)*(10*x-1)^2). - Colin Barker, Jan 27 2015
From Bernard Schott, Nov 20 2022: (Start)
a(n) = A033713(n) + 1.
a(n+1) = a(n) + 9 * A053541(n). (End)
MATHEMATICA
a[1] = 1; a[n_] := a[n] = 9*10^(n-2)*(n-1) + a[n-1]; Table[a[n], {n, 1, 17}] (* Jean-François Alcover, Jul 13 2012 *)
f[n_] := 1 + Sum[9 m*10^(m - 1), {m, n}]; Array[f, 18, 0] (* Robert G. Wilson v, Jun 29 2014 *)
LinearRecurrence[{21, -120, 100}, {1, 10, 190}, 20] (* Harvey P. Dale, Dec 03 2021 *)
PROG
(Magma) [(9*n*10^n-10*10^n+100)/90: n in [1..20]]; // Vincenzo Librandi, Jul 01 2014
(PARI) Vec(-x*(100*x^2-11*x+1)/((x-1)*(10*x-1)^2) + O(x^100)) \\ Colin Barker, Jan 27 2015
CROSSREFS
KEYWORD
nonn,base,nice,easy
AUTHOR
Olivier Gorin (gorin(AT)roazhon.inra.fr)
EXTENSIONS
More terms from Erich Friedman
STATUS
approved