A146160 - OEIS

%I #37 Dec 12 2023 08:23:25

%S 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 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 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 Period 4: repeat [1, 4, 1, 16].

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,1).

%F Continued fraction of (8 + sqrt(78))/14.

%F GCD(4k - k^2, 5k^2, 20k - 20k^2, 16 - 32k + 16k^2) for k = 1,2,3,...

%F From _Artur Jasinski_, Oct 29 2008: (Start)

%F a(n) = 1 when n congruent to 1 or 3 mod 4.

%F a(n) = 4 when n congruent to 2 mod 4.

%F a(n) = 16 when n congruent to 0 mod 4. (End)

%F From _Richard Choulet_, Nov 03 2008: (Start)

%F a(n+4) = a(n).

%F a(n) = (9/2)*(-1)^n + (11/2) + 6*cos(Pi*n/2).

%F O.g.f.: f(z) = a(0)+a(1)*z+... = (1+4*z+z^2+16*z^3)/(1-z^4). (End)

%F E.g.f.: sinh(x) + 20*(sinh(x/2))^2 - 12*(sin(x/2))^2. - _G. C. Greubel_, Feb 03 2016

%F a(n) = a(-n). - _Wesley Ivan Hurt_, Jun 15 2016

%F a(n) = A109008(n)^2. - _R. J. Mathar_, Feb 12 2019

%F From _Amiram Eldar_, Jan 01 2023: (Start)

%F Multiplicative with a(2) = 4, a(2^e) = 16 for e >= 2, and a(p^e) = 1 for p >= 3.

%F Dirichlet g.f.: zeta(s)*(12/4^s+3/2^s+1). (End)

%p A146160:=n->[1, 4, 1, 16][(n mod 4)+1]: seq(A146160(n), n=0..100); # _Wesley Ivan Hurt_, Jun 15 2016

%t Table[GCD[4k - k^2, 5k^2, 20k - 20k^2, 16 - 32k + 16k^2], {k, 100}]

%o (Magma) &cat[[1,4,1,16]^^20]; // _Vincenzo Librandi_, Feb 04 2016

%o (PARI) Vec((1+4*x+x^2+16*x^3)/(1-x^4) + O(x^100)) \\ _Altug Alkan_, Feb 04 2016

%Y Cf. A010156, A145996. [_Artur Jasinski_, Oct 29 2008]

%K nonn,easy,mult

%O 1,2

%A _Artur Jasinski_, Oct 27 2008

%E Choulet formula adapted for offset 1 from _Wesley Ivan Hurt_, Jun 15 2016