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

0,2

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

Table of n, a(n) for n=0..104.

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

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

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 *)

Prog

(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

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.

Sequence in context: A103585 A154595 A144437 * A353527 A220670 A264526

Adjacent sequences: A169606 A169607 A169608 * A169610 A169611 A169612

Keyword

easy,cofr,cons,nonn

Author

Klaus Brockhaus, Dec 03 2009

Extensions

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

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

Status

approved