A010686 Periodic sequence: repeat [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, 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, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• cons
• easy

Comments

Also continued fraction expansion of (5+3*sqrt(5))/10. - Bruno Berselli, Sep 30 2011

From Gary Detlefs, May 19 2014: (Start)

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.

1. If a = 5 mod 6 then c = 6.

2. If a = 5 mod 8 then c = 8.

3. If a = 5 mod 12 then c = 12.

4. If a = 5 mod 24 then c = 24.

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)

Decimal expansion of 5/33. - Stefano Spezia, Feb 09 2025

Examples

			0.15151515151515151515151515151515151515151...
		

Links

• Burkard Polster, Juggling, maths and a beautiful mind. [From Parthasarathy Nambi, Nov 20 2009]
• Vladimir Ivanovich Smirnov, A course of higher mathematics, vol. 1 , Pergamon Press, 1964, p. 2.
• Index entries for linear recurrences with constant coefficients, signature (0,1).

Crossrefs

Cf. A000364.

Programs

Maple

[seq (modp((4*n+1),8),n=0..80)]; # Zerinvary Lajos, Dec 01 2006

Mathematica

PadRight[{},120,{1,5}] (* Harvey P. Dale, Aug 19 2012 *)

Maxima

A010686(n):=if evenp(n) then 1 else 5$
makelist(A010686(n),n,0,30); /* Martin Ettl, Nov 09 2012 */

PARI

a(n)=n%2*4+1 \\ Charles R Greathouse IV, Sep 24 2015

Sage

[pow(5,n,12) for n in range(51)] # Zerinvary Lajos, Nov 25 2009

Formula

From Paul Barry, Jun 03 2003: (Start)

G.f.: (1+5*x)/((1-x)*(1+x)).

E.g.f.: 3*exp(x)-2*exp(-x).

a(n) = 3-2(-1)^n.

a(n) = 5^((1-(-1)^n)/2) = 5^(ceiling(n/2)-floor(n/2)). (End)

a(n) = 5^n mod 24. - Paul Curtz, Jan 09 2008

a(n) = 5^n mod 12. - Zerinvary Lajos, Nov 25 2009

a(n) = A000364(n+1) mod 10. - Paul Curtz, Feb 09 2010

a(n) = 11^n mod 6. - Vincenzo Librandi, Jun 01 2016

Extensions

Definition rewritten by Bruno Berselli, Sep 30 2011