A328683 Positive integers that are equal to 99...99 (repdigit with n digits 9) times the sum of their digits.

81, 1782, 26973, 359964, 4499955, 53999946, 629999937, 7199999928, 80999999919, 899999999910, 9899999999901, 107999999999892, 1169999999999883, 12599999999999874, 134999999999999865, 1439999999999999856, 15299999999999999847, 161999999999999999838

Offset

1,1

Comments

The idea of this sequence comes from a problem during the annual Moscow Mathematical Olympiad (MMO) in 2001 (see reference).

References

Roman Fedorov, Alexei Belov, Alexander Kovaldzhi, Ivan Yashchenko, Moscow Mathematical Olympiads, 2000-2005, Level B, Problem 5, 2001, MSRI, 2011, p. 8 and 70/71.

Links

Colin Barker, Table of n, a(n) for n = 1..995

Index to sequences related to Olympiads.

Index entries for linear recurrences with constant coefficients, signature (22,-141,220,-100).

Formula

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

From Colin Barker, Feb 25 2020: (Start)

G.f.: 81*x*(1 - 10*x^2) / ((1 - x)^2*(1 - 10*x)^2).

a(n) = 22*a(n-1) - 141*a(n-2) + 220*a(n-3) - 100*a(n-4) for n>4.

(End)

From Michel Marcus, Feb 25 2020: (Start)

a(n) = 9*A110807(n).

a(n) = n*A086580(n). (End)

Example

359964 = 36 * 9999 and the digital sum of 359964 = 36 , so 359964 = a(4).

Maple

C:=seq(9*n*(10^n-1), n=1..20);

Mathematica

Table[9*n*(10^n - 1), {n, 1, 18}] (* Amiram Eldar, Feb 25 2020 *)
LinearRecurrence[{22, -141, 220, -100}, {81, 1782, 26973, 359964}, 20] (* Harvey P. Dale, Feb 02 2025 *)

Prog

(PARI) Vec(81*x*(1 - 10*x^2) / ((1 - x)^2*(1 - 10*x)^2) + O(x^20)) \\ Colin Barker, Feb 25 2020

Crossrefs

Cf. A057147, A086580, A110807.

Sequence in context: A237091 A253306 A237801 * A236057 A205910 A236811

Adjacent sequences: A328680 A328681 A328682 * A328684 A328685 A328686

Keyword

nonn,base

Author

Bernard Schott, Feb 25 2020

Status

approved