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%I #33 Dec 12 2023 07:46:28
%S 1,3,3,3,1,3,3,3,1,3,3,3,1,3,3,3,1,3,3,3,1,3,3,3,1,3,3,3,1,3,3,3,1,3,
%T 3,3,1,3,3,3,1,3,3,3,1,3,3,3,1,3,3,3,1,3,3,3,1,3,3,3,1,3,3,3,1,3,3,3,
%U 1,3,3,3,1,3,3,3,1,3,3,3,1,3,3,3,1,3,3,3,1,3,3,3,1,3,3,3,1,3,3,3,1,3,3,3,1
%N Period 4: repeat [1, 3, 3, 3].
%C Denominator of x(n) = x(n-1) + x(n-2), x(0)=0, x(1)=1/3; numerator = A167816(n).
%C Continued fraction expansion of (33 + sqrt(2805))/66. - _Klaus Brockhaus_, May 06 2010
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,1).
%F a(n) = 3 - 2 * 0^(n mod 4).
%F G.f.: (1 + 3*x + 3*x^2 + 3*x^3)/(1-x^4). - _Klaus Brockhaus_, May 06 2010
%F a(n) = 5/2 - cos(Pi*n/2) - (-1)^n/2. - _R. J. Mathar_, Oct 08 2011
%F E.g.f.: -cos(x) + 3*sinh(x) + 2*cosh(x). - _Ilya Gutkovskiy_, Jun 27 2016
%F a(n) = a(n-4) for n>3. - _Wesley Ivan Hurt_, Jul 09 2016
%p seq(op([1, 3, 3, 3]), n=0..50); # _Wesley Ivan Hurt_, Jul 09 2016
%t Denominator[LinearRecurrence[{1,1},{0,1/3},110]] (* or *) PadRight[{},110,{1,3,3,3}] (* _Harvey P. Dale_, Dec 07 2014 *)
%t LinearRecurrence[{0, 0, 0, 1},{1, 3, 3, 3},105] (* _Ray Chandler_, Aug 03 2015 *)
%o (Magma) &cat[[1, 3, 3, 3]: n in [0..50]]; // _Vincenzo Librandi_, Dec 28 2010
%Y Cf. A130196.
%Y Cf. A177344 (decimal expansion of (33+sqrt(2805))/66). - _Klaus Brockhaus_, May 06 2010
%K frac,nonn,easy
%O 0,2
%A _Reinhard Zumkeller_, Nov 13 2009
%E Definition corrected by _D. S. McNeil_, May 09 2010
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