A010684 Period 2: repeat (1,3); offset 0.

1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• cons
• easy

Comments

Hankel transform is [1,-8,0,0,0,0,0,0,0,0,...]. - Philippe Deléham, Mar 29 2007

Binomial transform gives [1,4,8,16,32,64,...] (A151821(n+1)). - Philippe Deléham, Sep 17 2009

Continued fraction expansion of (3+sqrt(21))/6. - Klaus Brockhaus, May 04 2010

Positive sum of the coordinates from the image of the point (1,-2) after n 90-degree rotations about the origin. - Wesley Ivan Hurt, Jul 06 2013

This sequence can be generated by an infinite number of formulas having the form a^(b*n) mod c where a is congruent to 3 mod 4 and b is any odd number. If a is congruent to 3 mod 4 then c can be 4; if a is also congruent to 3 mod 8 then c can be 8. For example: a(n)= 15^(3*n) mod 4, a(n) = 19^(5*n) mod 4, a(n) = 19^(5*n) mod 8. - Gary Detlefs, May 19 2014

This sequence is also the unsigned periodic Schick sequence for p = 5. See the Schick reference, p. 158, for p = 5.- Wolfdieter Lang, Apr 03 2020

Digits following the decimal point when 1/3 is converted to base 5. - Jamie Robert Creasey, Oct 15 2021

Decimal expansion of 13/99. - Stefano Spezia, Feb 09 2025

Examples

			0.131313131313131313131313131313131313131313131...
		

References

• Carl Schick, Trigonometrie und unterhaltsame Zahlentheorie, Bokos Druck, Zürich, 2003 (ISBN 3-9522917-0-6). Tables 3.1 to 3.10, for odd p = 3..113 (with gaps), pp. 158-166.

Links

• Index entries for linear recurrences with constant coefficients, signature (0,1).

Crossrefs

Cf. A112030, A112033, A176014 (decimal expansion of (3+sqrt(21))/6).

Programs

Maple

[seq (modp((2*n+1),4),n=0..80)]; # Zerinvary Lajos, Nov 30 2006

Mathematica

Table[2-(-1)^n, {n, 0, 100}] (* Wesley Ivan Hurt, Mar 24 2014 *)
PadRight[{},120,{1,3}] (* Harvey P. Dale, Mar 11 2025 *)

PARI

a(n)=1+n%2*2 \\ Charles R Greathouse IV, Dec 28 2011

Python

def A010684(n): return 3 if n&1 else 1 # Chai Wah Wu, Jan 17 2023

Sage

[power_mod(3, n, 8)for n in range(0, 81)] # Zerinvary Lajos, Nov 24 2009

Formula

From Paul Barry, Apr 29 2003: (Start)

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

G.f.: (1+3x)/((1-x)(1+x)).

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

a(n) = 2*A153643(n) - A153643(n+1). - Paul Curtz, Dec 30 2008

a(n) = 3^(n mod 2). - Jaume Oliver Lafont, Mar 27 2009

a(n) = 7^n mod 4. - Vincenzo Librandi, Feb 07 2011

a(n) = 1 + 2*(n mod 2). - Wesley Ivan Hurt, Jul 06 2013

a(n) = A000034(n) + A000035(n). - James Spahlinger, Feb 14 2016