A220094 Sum of the n-digit base-ten numbers whose digits are nonzero.

45, 4455, 404595, 36446355, 3280467195, 295244704755, 26572047342795, 2391484476085155, 215233604784766395, 19371024448062897555, 1743392200482566077995, 156905298044843094701955, 14121476824048587852317595, 1270932914164487290670858355

Offset

1,1

Comments

For n >= 1, a(n) is the sum of the numbers with n digits in base ten whose digits belong to the set {1,2,3,4,5,6,7,8,9}.

If E_n is the set of the numbers with n digits in base ten whose digits belong to {1,2,3,4,5,6,7,8,9}, then card(E_n) = 9^n (see A001019).

References

A. Ducos, Eléments fondamentaux de Math Sup, Ellipses, 1994, exercice 9, p. 126.

Links

Table of n, a(n) for n=1..14.

Bernard Schott and Raymond Cordier, Question Comtet 16 (French mathematical forum les-mathematiques.net)

Index entries for linear recurrences with constant coefficients, signature (99,-810).

Formula

a(n) = 5*9^(n-1)*(10^n-1).

Generalization to base b with n-digit numbers whose digits belong to {1,2,...,b-1}: a_b(n) = (b/2)*(b-1)^(n-1)*(b^n-1).

From Colin Barker, Jan 04 2013: (Start)

a(n) = 99*a(n-1) - 810*a(n-2).

G.f.: 45*x/((9*x-1)*(90*x-1)). (End)

Example

For n=2, in base ten, a(2) = 11+12+...+19+21+...+89+91+...+98+99 = 4455.

Maple

:= n->5*9^(n-1)*(10^n-1);

Mathematica

Table[5*9^(n - 1)*(10^n - 1), {n, 20}] (* T. D. Noe, Dec 31 2012 *)

Prog

(PARI) a(n)=5*9^(n-1)*(10^n-1) \\ Charles R Greathouse IV, Jul 06 2017

Crossrefs

Cf. A192107, A192370.

Sequence in context: A113630 A143004 A004707 * A274603 A036521 A328356

Adjacent sequences: A220091 A220092 A220093 * A220095 A220096 A220097

Keyword

nonn,base,easy

Author

Bernard Schott, Dec 04 2012

Status

approved