cp's OEIS Frontend

A010685 Period 2: repeat (1,4).

%I A010685 #69 Dec 14 2023 06:07:14
%S A010685 1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,
%T A010685 1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,
%U A010685 1,4,1,4,1,4,1,4,1,4,1,4,1
%N A010685 Period 2: repeat (1,4).
%C A010685 Continued fraction of (1 + sqrt(2))/2. - _R. J. Mathar_, Nov 21 2011
%C A010685 This sequence can be generated by an infinite number of formulas all having the form a^(b*n) mod c subject to the following conditions. The number a can be congruent to either 2,3, or 4 mod 5 (A047202). If a is congruent to 2 or 3 mod 5, then b can be any number of the form 4k+2 and c = 5 or 15. If a is congruent to 4 mod 5, then b can be any number of the form  2k+1 and c = 5.  For example: a(n) = 29^(13*n) mod 5,  a(n) = 24^(11*n) mod 5, and a(n) = 22^(10*n) mod 15. - _Gary Detlefs_, May 19 2014
%H A010685 Matthew House, <a href="/A010685/b010685.txt">Table of n, a(n) for n = 0..1000</a>
%H A010685 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (0,1).
%F A010685 a(2n) = 1, a(2n+1) = 4.
%F A010685 From _Paul Barry_, Jun 03 2003: (Start)
%F A010685 G.f.: (1+4*x)/((1-x)*(1+x)).
%F A010685 E.g.f.: (5*exp(x) - 3*exp(-x))/2.
%F A010685 a(n) = (5 - 3*(-1)^n)/2.
%F A010685 a(n) = 4^((1-(-1)^n)/2) = 2^(1-(-1)^n) = 2/(2^((-1)^n)).
%F A010685 a(n) = 4^(ceiling(n/2) - floor(n/2)). (End)
%F A010685 a(n) = gcd((n-1)^2, (n+1)^2). - _Paul Barry_, Sep 16 2004
%F A010685 a(n) = A160700(A000302(n)). - _Reinhard Zumkeller_, Jun 10 2009
%F A010685 a(n) = 4^n mod 5. - _Zerinvary Lajos_, Nov 26 2009
%F A010685 a(n) = 4^(n mod 2). - _Wesley Ivan Hurt_, Mar 29 2014
%p A010685 A010685 := proc(n)
%p A010685     if type(n,'even') then
%p A010685         1 ;
%p A010685     else
%p A010685         4;
%p A010685     end if;
%p A010685 end proc: # _R. J. Mathar_, Aug 03 2015
%t A010685 Table[(5-3(-1)^n)/2, {n, 0, 100}] (* _Wesley Ivan Hurt_, Mar 26 2014 *)
%t A010685 PadRight[{},120,{1,4}] (* _Harvey P. Dale_, Aug 08 2022 *)
%o A010685 (Sage) [power_mod(4,n,5)for n in range(0,81)] # _Zerinvary Lajos_, Nov 26 2009
%o A010685 (PARI) values(m)=my(v=[]);for(i=1,m,v=concat([1,4],v));v; /* _Anders Hellström_, Aug 03 2015 */
%o A010685 (Magma) [Modexp(4,n,5): n in [0..100]]; // _G. C. Greubel_, Nov 22 2021
%Y A010685 Cf. sequences listed in Comments section of A283393.
%Y A010685 Cf. A047202.
%K A010685 nonn,easy
%O A010685 0,2
%A A010685 _N. J. A. Sloane_