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A010686 Periodic sequence: repeat [1, 5].

%I A010686 #72 Feb 10 2025 02:20:46
%S A010686 1,5,1,5,1,5,1,5,1,5,1,5,1,5,1,5,1,5,1,5,1,5,1,5,1,5,1,5,1,5,1,5,1,5,
%T A010686 1,5,1,5,1,5,1,5,1,5,1,5,1,5,1,5,1,5,1,5,1,5,1,5,1,5,1,5,1,5,1,5,1,5,
%U A010686 1,5,1,5,1,5,1,5,1,5,1,5,1
%N A010686 Periodic sequence: repeat [1, 5].
%C A010686 Also continued fraction expansion of (5+3*sqrt(5))/10. - _Bruno Berselli_, Sep 30 2011
%C A010686 From _Gary Detlefs_, May 19 2014: (Start)
%C A010686 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 is congruent to either 5,11,13,17,21, or 23 mod 24 and b is of the form 2k+1.
%C A010686   1. If a = 5 mod 6 then c = 6.
%C A010686   2. If a = 5 mod 8 then c = 8.
%C A010686   3. If a = 5 mod 12 then c = 12.
%C A010686   4. If a = 5 mod 24 then c = 24.
%C A010686 For example: a(n)= 13^(5*n) mod 8, a(n)= 29^(7*n) mod c where c is any number in {6,8,12,24}. (End)
%C A010686 Decimal expansion of 5/33. - _Stefano Spezia_, Feb 09 2025
%H A010686 Burkard Polster, <a href="http://plus.maths.org/issue52/features/polster/index.html">Juggling, maths and a beautiful mind</a>. [From Parthasarathy Nambi, Nov 20 2009]
%H A010686 Vladimir Ivanovich Smirnov, <a href="https://archive.org/details/v-i-smirnov.-a-course-of-higher-mathematics-vol-1/page/1/mode/2up">A course of higher mathematics</a>, vol. 1 , Pergamon Press, 1964, p. 2.
%H A010686 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (0,1).
%F A010686 From _Paul Barry_, Jun 03 2003: (Start)
%F A010686 G.f.: (1+5*x)/((1-x)*(1+x)).
%F A010686 E.g.f.: 3*exp(x)-2*exp(-x).
%F A010686 a(n) = 3-2(-1)^n.
%F A010686 a(n) = 5^((1-(-1)^n)/2) = 5^(ceiling(n/2)-floor(n/2)). (End)
%F A010686 a(n) = 5^n mod 24. - _Paul Curtz_, Jan 09 2008
%F A010686 a(n) = 5^n mod 12. - _Zerinvary Lajos_, Nov 25 2009
%F A010686 a(n) = A000364(n+1) mod 10. - _Paul Curtz_, Feb 09 2010
%F A010686 a(n) = 11^n mod 6. - _Vincenzo Librandi_, Jun 01 2016
%e A010686 0.15151515151515151515151515151515151515151...
%p A010686 [seq (modp((4*n+1),8),n=0..80)]; # _Zerinvary Lajos_, Dec 01 2006
%t A010686 PadRight[{},120,{1,5}] (* _Harvey P. Dale_, Aug 19 2012 *)
%o A010686 (Sage) [pow(5,n,12) for n in range(51)] # _Zerinvary Lajos_, Nov 25 2009
%o A010686 (Maxima) A010686(n):=if evenp(n) then 1 else 5$
%o A010686 makelist(A010686(n),n,0,30); /* _Martin Ettl_, Nov 09 2012 */
%o A010686 (PARI) a(n)=n%2*4+1 \\ _Charles R Greathouse IV_, Sep 24 2015
%Y A010686 Cf. A000364.
%K A010686 nonn,cons,easy
%O A010686 0,2
%A A010686 _N. J. A. Sloane_
%E A010686 Definition rewritten by _Bruno Berselli_, Sep 30 2011