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A021115 Decimal expansion of 1/111.

%I A021115 #49 Sep 08 2022 08:44:45
%S A021115 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,
%T A021115 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,
%U A021115 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
%N A021115 Decimal expansion of 1/111.
%C A021115 Period 3: repeat [0, 0, 9]. - _Joerg Arndt_, Sep 29 2015
%H A021115 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,1).
%F A021115 From _Alexander R. Povolotsky_, Sep 29 2015: (Start)
%F A021115 G.f.: 9*x^2/(1 - x^3).
%F A021115 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_.
%F A021115 a(n) = 3*( 1 + cos(2*(n+1)*Pi/3) + cos(4*(n+1)*Pi/3) ).
%F A021115 a(n) = a(n + 3) for n>2.
%F A021115 a(n) = A056992(n+1) - (3*(n+1)^4+3*(n+1)^6+4*(n+1)^8) mod 9. (End)
%F A021115 a(n) = 9*A022003(n). - _Robert Israel_, Sep 30 2015
%p A021115 seq(op([0,0,9]), i=1..100); # _Robert Israel_, Sep 30 2015
%t A021115 PadRight[{}, 100, {0, 0, 9}] (* _Wesley Ivan Hurt_, Jun 30 2016 *)
%o A021115 (PARI) 1/111.0 \\ _Michel Marcus_, Sep 29 2015
%o A021115 (PARI) a(n) = 9*(n*(n-1)%3)/2;
%o A021115 vector(100, n, a(n-1)) \\ _Altug Alkan_, Sep 30 2015
%o A021115 (Magma) &cat[[0,0,9]: n in [0..35]]; // _Vincenzo Librandi_, Sep 29 2015
%Y A021115 Cf. A022003, A056992.
%K A021115 nonn,cons,easy
%O A021115 0,3
%A A021115 _N. J. A. Sloane_
%E A021115 Edited by _Bruno Berselli_, Dec 15 2015