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