A305322 Repdigit numbers that are divisible by 3.

0, 3, 6, 9, 33, 66, 99, 111, 222, 333, 444, 555, 666, 777, 888, 999, 3333, 6666, 9999, 33333, 66666, 99999, 111111, 222222, 333333, 444444, 555555, 666666, 777777, 888888, 999999, 3333333, 6666666, 9999999, 33333333, 66666666, 99999999, 111111111, 222222222

Offset

1,2

Comments

The terms > 0 are (10^d-1)*k/9 for k=1..9 if d is divisible by 3, and for k=3,6,9 otherwise. - Robert Israel, Jun 01 2018

Repdigit remainders A010785(k) mod 3 have period 27. - Karl-Heinz Hofmann, Nov 11 2023

Links

Robert Israel, Table of n, a(n) for n = 1..4996

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,1001,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1000).

Formula

From Alois P. Heinz, May 30 2018: (Start)

{ A008585 } intersect { A010785 }.

G.f.: 3*(300*x^20 + 200*x^19 + 100*x^18 + 330*x^17 + 220*x^16 + 110*x^15 + 333*x^14 + 296*x^13 + 259*x^12 + 222*x^11 + 185*x^10 + 148*x^9 + 111*x^8 + 74*x^7 + 37*x^6 + 33*x^5 + 22*x^4 + 11*x^3 + 3*x^2 + 2*x + 1)*x^2 / ((x-1) *(x^2 + x + 1) *(x^4 + x^3 + x^2 + x + 1) *(10*x^5-1) *(x^8 - x^7 + x^5 - x^4 + x^3 - x + 1) *(100*x^10 + 10*x^5 + 1)).

a(n) = 1001*a(n-15) - 1000*a(n-30). (End)

From Karl-Heinz Hofmann, Nov 11 2023: (Start)

a(n) = A010785(floor((n-1)/15)*27 + ((n-1) mod 15)*3) iff (n-1 <= 6 (mod 15)).

a(n) = A010785(floor((n-1)/15)*27 + ((n-1) mod 15) + 12) iff (n-1 > 6 (mod 15)).

(End)

Example

111 / 3 = 37;
222 / 3 = 74;
333 / 3 = 111;
444 / 3 = 148;
555 / 3 = 185.

Maple

L:= proc(d) if d mod 3 = 0 then [$1..9] else [3, 6, 9] fi end proc:
0, seq(seq((10^d-1)/9*k, k=L(d)), d=1..9); # Robert Israel, Jun 01 2018

Prog

(Python)
def A010785(n): return (n - 9*((n-1)//9))*(10**((n+8)//9) - 1)//9
def A305322(n):
    d0, d1 = divmod(n-1, 15)
    if d1 < 7: return A010785(d0 * 27 + d1 * 3)
    return A010785(d0 * 27 + d1 + 12) # Karl-Heinz Hofmann, Nov 26 2023

Crossrefs

Cf. A008585, A010785.

Cf. A002279 (divisor 5), A366596 (divisor 7), A083118 (the impossible divisors).

Sequence in context: A254616 A195205 A045638 * A038224 A133195 A196156

Adjacent sequences: A305319 A305320 A305321 * A305323 A305324 A305325

Keyword

nonn,base,easy

Author

Kritsada Moomuang, May 30 2018

Extensions

Name clarified by Felix Fröhlich, Jun 01 2018

Status

approved