A099676 Partial sums of repdigits of A002283.

9, 108, 1107, 11106, 111105, 1111104, 11111103, 111111102, 1111111101, 11111111100, 111111111099, 1111111111098, 11111111111097, 111111111111096, 1111111111111095, 11111111111111094, 111111111111111093, 1111111111111111092, 11111111111111111091

Offset

1,1

Comments

a(n) is the maximal positive integer k such that the sequence 1, 2, 3, 4, ..., k-1, k has a total of n*k digits. - Bui Quang Tuan, Mar 12 2015

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (12,-21,10).

Formula

a(n) = (10/9)*(10^n-1) - n. - R. Piyo (nagoya314(AT)yahoo.com), Dec 10 2004

From Colin Barker, Mar 12 2014: (Start)

a(n) = 12*a(n-1)-21*a(n-2)+10*a(n-3).

G.f.: -9*x / ((x-1)^2*(10*x-1)). (End)

E.g.f.: exp(x)*(10*(exp(9*x) - 1) - 9*x)/9. - Stefano Spezia, Sep 13 2023

Example

9 + 99 + 999 + 9999 + 99999 = a(5) = 111105.

Maple

a:=n->sum((10^(n-j)-1^(n-j)), j=0..n): seq(a(n), n=1..17); # Zerinvary Lajos, Jan 15 2007

Mathematica

<<NumberTheory`NumberTheoryFunctions` Table[{k, Table[Apply[Plus, Table[k*(10^n-1)/9, {n, 1, m}]], {m, 1, 35}]}, {k, 1, 9}]
CoefficientList[Series[-9/((x - 1)^2 (10 x - 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Mar 14 2014 *)
LinearRecurrence[{12, -21, 10}, {9, 108, 1107}, 20] (* Harvey P. Dale, Apr 18 2015 *)

Prog

(Sage) [gaussian_binomial(n, 1, 10)-n for n in range(2, 19)] # Zerinvary Lajos, May 29 2009
(PARI) Vec(-9*x/((x-1)^2*(10*x-1)) + O(x^100)) \\ Colin Barker, Mar 12 2014
(Magma) [(10/9)*(10^n-1)-n: n in [1..20]]; // Vincenzo Librandi, Mar 14 2014

Crossrefs

Cf. A057932, A002275-A002283, A099669-A099674.

Sequence in context: A249048 A080505 A104224 * A268839 A214668 A234467

Adjacent sequences: A099673 A099674 A099675 * A099677 A099678 A099679

Keyword

base,nonn,easy

Author

Labos Elemer, Nov 17 2004

Status

approved