A169609 Period 3: repeat [1, 3, 3].

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

Offset: 0

Author

Klaus Brockhaus, Dec 03 2009

Keywords

• easy
• cofr
• cons
• nonn

Comments

Interleaving of A000012, A010701 and A010701.

Also continued fraction expansion of (5+sqrt(65))/10 = 1.3062257748....

Also decimal expansion of 133/999.

a(n) = A144437(n) for n > 0.

Unsigned version of A154595.

Binomial transform of A168615.

Inverse binomial transform of A168673.

Essentially first differences of A047347.

Links

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

Crossrefs

Cf. A000012 (all 1's sequence), A010701 (all 3's sequence), A144437 (repeat 3, 3, 1), A154595 (repeat 1, 3, 3, -1, -3, -3), A168615, A168673, A047347 (congruent to {0, 1, 4} mod 7), A010684 (repeat 1, 3).

Cf. A171419 (decimal expansion of (5+sqrt(65))/10).

Cf. A146094.

Programs

Magma

[ n mod 3 eq 0 select 1 else 3: n in [0..104] ];

Magma

&cat [[1, 3, 3]^^30]; // Wesley Ivan Hurt, Jul 02 2016

Maple

seq(op([1, 3, 3]), n=0..50); # Wesley Ivan Hurt, Jul 02 2016

Mathematica

PadRight[{},120,{1,3,3}] (* or *) LinearRecurrence[{0,0,1},{1,3,3},120] (* Harvey P. Dale, Apr 29 2015 *)

Formula

a(n) = a(n-3) for n > 2, with a(0) = 1, a(1) = 3, a(2) = 3.

G.f.: (1+3*x+3*x^2)/(1-x^3).

a(n) = (7/3)+(2/3)*cos((2*Pi/3)*(n+1))-(2*sqrt(3)/3)*sin((2*Pi/3)*(n+1)). [Richard Choulet, Mar 15 2010]

a(n) = a(n-a(n-2)) for n>=2. Example: a(5) = a(5-a(3)) = a(5-a(3-a(1))) = a(5-a(3-3)) = a(5-a(0)) = a(5-1) = a(4) = a(4-a(2)) = a(4-3) = a(1) = 3. [Richard Choulet, Mar 15 2010; edited by Klaus Brockhaus, Nov 21 2010]

a(n) = 1 + 2*sgn(n mod 3). - Wesley Ivan Hurt, Jul 02 2016

a(n) = 3/gcd(n,3). - Wesley Ivan Hurt, Jul 11 2016

Extensions

Keywords cofr, cons added by Klaus Brockhaus, Apr 20 2010

Minor edits, crossref added by Klaus Brockhaus, May 03 2010