A021115 Decimal expansion of 1/111.

0, 0, 9, 0, 0, 9, 0, 0, 9, 0, 0, 9, 0, 0, 9, 0, 0, 9, 0, 0, 9, 0, 0, 9, 0, 0, 9, 0, 0, 9, 0, 0, 9, 0, 0, 9, 0, 0, 9, 0, 0, 9, 0, 0, 9, 0, 0, 9, 0, 0, 9, 0, 0, 9, 0, 0, 9, 0, 0, 9, 0, 0, 9, 0, 0, 9, 0, 0, 9, 0, 0, 9, 0, 0, 9, 0, 0, 9, 0, 0, 9, 0, 0, 9, 0, 0, 9, 0, 0, 9, 0, 0, 9, 0, 0, 9, 0, 0, 9

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• cons
• easy

Comments

Period 3: repeat [0, 0, 9]. - Joerg Arndt, Sep 29 2015

Links

• Index entries for linear recurrences with constant coefficients, signature (0,0,1).

Crossrefs

Cf. A022003, A056992.

Programs

Magma

&cat[[0,0,9]: n in [0..35]]; // Vincenzo Librandi, Sep 29 2015

Maple

seq(op([0,0,9]), i=1..100); # Robert Israel, Sep 30 2015

Mathematica

PadRight[{}, 100, {0, 0, 9}] (* Wesley Ivan Hurt, Jun 30 2016 *)

PARI

1/111.0 \\ Michel Marcus, Sep 29 2015

PARI

a(n) = 9*(n*(n-1)%3)/2;
vector(100, n, a(n-1)) \\ Altug Alkan, Sep 30 2015

Formula

From Alexander R. Povolotsky, Sep 29 2015: (Start)

G.f.: 9*x^2/(1 - x^3).

a(n) = (9/2)*( (n-1)*n mod 3 ) = 4*(n mod 3) - 2*((n+1) mod 3) + ((n+2) mod 3), formulas suggested by Giovanni Resta.

a(n) = 3*( 1 + cos(2*(n+1)*Pi/3) + cos(4*(n+1)*Pi/3) ).

a(n) = a(n + 3) for n>2.

a(n) = A056992(n+1) - (3*(n+1)^4+3*(n+1)^6+4*(n+1)^8) mod 9. (End)

a(n) = 9*A022003(n). - Robert Israel, Sep 30 2015

Extensions

Edited by Bruno Berselli, Dec 15 2015