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A305322
Repdigit numbers that are divisible by 3.
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
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. A002279 (divisor 5), A366596 (divisor 7), A083118 (the impossible divisors).
Sequence in context: A254616 A195205 A045638 * A038224 A133195 A196156
KEYWORD
nonn,base,easy
AUTHOR
Kritsada Moomuang, May 30 2018
EXTENSIONS
Name clarified by Felix Fröhlich, Jun 01 2018
STATUS
approved