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%I #36 Dec 12 2023 08:10:03
%S 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,
%T 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,
%U 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
%N Period 3: repeat [1, 3, 3].
%C Interleaving of A000012, A010701 and A010701.
%C Also continued fraction expansion of (5+sqrt(65))/10 = 1.3062257748....
%C Also decimal expansion of 133/999.
%C a(n) = A144437(n) for n > 0.
%C Unsigned version of A154595.
%C Binomial transform of A168615.
%C Inverse binomial transform of A168673.
%C Essentially first differences of A047347.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,1).
%F a(n) = a(n-3) for n > 2, with a(0) = 1, a(1) = 3, a(2) = 3.
%F G.f.: (1+3*x+3*x^2)/(1-x^3).
%F 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]
%F 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]
%F a(n) = 1 + 2*sgn(n mod 3). - _Wesley Ivan Hurt_, Jul 02 2016
%F a(n) = 3/gcd(n,3). - _Wesley Ivan Hurt_, Jul 11 2016
%p seq(op([1, 3, 3]), n=0..50); # _Wesley Ivan Hurt_, Jul 02 2016
%t PadRight[{},120,{1,3,3}] (* or *) LinearRecurrence[{0,0,1},{1,3,3},120] (* _Harvey P. Dale_, Apr 29 2015 *)
%o (Magma) [ n mod 3 eq 0 select 1 else 3: n in [0..104] ];
%o (Magma) &cat [[1, 3, 3]^^30]; // _Wesley Ivan Hurt_, Jul 02 2016
%Y 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).
%Y Cf. A171419 (decimal expansion of (5+sqrt(65))/10).
%Y Cf. A146094.
%K easy,cofr,cons,nonn
%O 0,2
%A _Klaus Brockhaus_, Dec 03 2009
%E Keywords cofr, cons added by _Klaus Brockhaus_, Apr 20 2010
%E Minor edits, crossref added by _Klaus Brockhaus_, May 03 2010
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