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A146160 Period 4: repeat [1, 4, 1, 16].

%I A146160 #39 Mar 29 2025 15:14:48
%S A146160 1,4,1,16,1,4,1,16,1,4,1,16,1,4,1,16,1,4,1,16,1,4,1,16,1,4,1,16,1,4,1,
%T A146160 16,1,4,1,16,1,4,1,16,1,4,1,16,1,4,1,16,1,4,1,16,1,4,1,16,1,4,1,16,1,
%U A146160 4,1,16,1,4,1,16,1,4,1,16,1,4,1,16,1,4,1,16,1,4,1,16,1,4,1,16,1,4,1,16,1
%N A146160 Period 4: repeat [1, 4, 1, 16].
%H A146160 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,1).
%F A146160 Continued fraction of (8 + sqrt(78))/14.
%F A146160 GCD(4k - k^2, 5k^2, 20k - 20k^2, 16 - 32k + 16k^2) for k = 1,2,3,...
%F A146160 From _Artur Jasinski_, Oct 29 2008: (Start)
%F A146160 a(n) = 1 when n congruent to 1 or 3 mod 4.
%F A146160 a(n) = 4 when n congruent to 2 mod 4.
%F A146160 a(n) = 16 when n congruent to 0 mod 4. (End)
%F A146160 From _Richard Choulet_, Nov 03 2008: (Start)
%F A146160 a(n+4) = a(n).
%F A146160 a(n) = (9/2)*(-1)^n + (11/2) + 6*cos(Pi*n/2).
%F A146160 O.g.f.: f(z) = a(0)+a(1)*z+... = (1+4*z+z^2+16*z^3)/(1-z^4). (End)
%F A146160 E.g.f.: sinh(x) + 20*(sinh(x/2))^2 - 12*(sin(x/2))^2. - _G. C. Greubel_, Feb 03 2016
%F A146160 a(n) = a(-n). - _Wesley Ivan Hurt_, Jun 15 2016
%F A146160 a(n) = A109008(n)^2. - _R. J. Mathar_, Feb 12 2019
%F A146160 From _Amiram Eldar_, Jan 01 2023: (Start)
%F A146160 Multiplicative with a(2) = 4, a(2^e) = 16 for e >= 2, and a(p^e) = 1 for p >= 3.
%F A146160 Dirichlet g.f.: zeta(s)*(12/4^s+3/2^s+1). (End)
%p A146160 A146160:=n->[1, 4, 1, 16][(n mod 4)+1]: seq(A146160(n), n=0..100); # _Wesley Ivan Hurt_, Jun 15 2016
%t A146160 Table[GCD[4k - k^2, 5k^2, 20k - 20k^2, 16 - 32k + 16k^2], {k, 100}]
%t A146160 PadRight[{},100,{1,4,1,16}] (* or *) LinearRecurrence[{0,0,0,1},{1,4,1,16},90] (* _Harvey P. Dale_, Mar 29 2025 *)
%o A146160 (Magma) &cat[[1,4,1,16]^^20]; // _Vincenzo Librandi_, Feb 04 2016
%o A146160 (PARI) Vec((1+4*x+x^2+16*x^3)/(1-x^4) + O(x^100)) \\ _Altug Alkan_, Feb 04 2016
%Y A146160 Cf. A010156, A145996. [_Artur Jasinski_, Oct 29 2008]
%K A146160 nonn,easy,mult
%O A146160 1,2
%A A146160 _Artur Jasinski_, Oct 27 2008
%E A146160 Choulet formula adapted for offset 1 from _Wesley Ivan Hurt_, Jun 15 2016