%I #75 Dec 14 2023 06:07:07

%S 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,

%T 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,

%U 1,3,1,3,1,3,1,3,1,3,1,3,1

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

%C Hankel transform is [1,-8,0,0,0,0,0,0,0,0,...]. - _Philippe Deléham_, Mar 29 2007

%C Binomial transform gives [1,4,8,16,32,64,...] (A151821(n+1)). - _Philippe Deléham_, Sep 17 2009

%C Continued fraction expansion of (3+sqrt(21))/6. - _Klaus Brockhaus_, May 04 2010

%C 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

%C 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

%C 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

%C Digits following the decimal point when 1/3 is converted to base 5. - _Jamie Robert Creasey_, Oct 15 2021

%D 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.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (0,1).

%F From _Paul Barry_, Apr 29 2003: (Start)

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

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

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

%F a(n) = 2*A153643(n) - A153643(n+1). - _Paul Curtz_, Dec 30 2008

%F a(n) = 3^(n mod 2). - _Jaume Oliver Lafont_, Mar 27 2009

%F a(n) = 7^n mod 4. - _Vincenzo Librandi_, Feb 07 2011

%F a(n) = 1 + 2*(n mod 2). - _Wesley Ivan Hurt_, Jul 06 2013

%F a(n) = A000034(n) + A000035(n). - _James Spahlinger_, Feb 14 2016

%p [seq (modp((2*n+1),4),n=0..80)]; # _Zerinvary Lajos_, Nov 30 2006

%t Table[2-(-1)^n, {n, 0, 100}] (* _Wesley Ivan Hurt_, Mar 24 2014 *)

%o (Sage) [power_mod(3, n, 8)for n in range(0, 81)] # _Zerinvary Lajos_, Nov 24 2009

%o (PARI) a(n)=1+n%2*2 \\ _Charles R Greathouse IV_, Dec 28 2011

%o (Python)

%o def A010684(n): return 3 if n&1 else 1 # _Chai Wah Wu_, Jan 17 2023

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

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_