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A070431 a(n) = n^2 mod 6.

%I A070431 #51 Dec 27 2023 14:32:43
%S A070431 0,1,4,3,4,1,0,1,4,3,4,1,0,1,4,3,4,1,0,1,4,3,4,1,0,1,4,3,4,1,0,1,4,3,
%T A070431 4,1,0,1,4,3,4,1,0,1,4,3,4,1,0,1,4,3,4,1,0,1,4,3,4,1,0,1,4,3,4,1,0,1,
%U A070431 4,3,4,1,0,1,4,3,4,1,0,1,4,3,4,1,0,1,4,3,4,1,0,1,4,3,4,1,0,1,4,3,4
%N A070431 a(n) = n^2 mod 6.
%C A070431 a(m*n) = a(m)*a(n) mod 6; a(3*n+k) = a(3*n-k) for k <= 3*n. - _Reinhard Zumkeller_, Apr 24 2009
%C A070431 Equivalently n^6 mod 6. - _Zerinvary Lajos_, Nov 06 2009
%C A070431 Equivalently: n^(2*m + 4) mod 6; n^(2*m + 2) mod 6. - _G. C. Greubel_, Apr 01 2016
%H A070431 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0, 0, 0, 0, 0, 1).
%F A070431 G.f.: -x*(1+4*x+3*x^2+4*x^3+x^4)/((x-1)*(1+x)*(1+x+x^2)*(x^2-x+1)). - _R. J. Mathar_, Jul 23 2009
%F A070431 a(n) = a(n-6). - _Reinhard Zumkeller_, Apr 24 2009
%F A070431 From _G. C. Greubel_, Apr 01 2016: (Start)
%F A070431 a(6*m) = 0.
%F A070431 a(2*n) = 4*A011655(n).
%F A070431 a(n) = (1/6)*(13 + 3*(-1)^n - 12*cos(n*Pi/3) - 4*cos(2*n*Pi/3)).
%F A070431 G.f.: (x +4*x^2 +3*x^3 + 4*x^4 +x^5)/(1 - x^6). (End)
%p A070431 A070431:=n->n^2 mod 6: seq(A070431(n), n=0..100); # _Wesley Ivan Hurt_, Apr 01 2016
%t A070431 Table[Mod[n^2, 6], {n, 0, 200}] (* _Vladimir Joseph Stephan Orlovsky_, Apr 21 2011 *)
%t A070431 LinearRecurrence[{0, 0, 0, 0, 0, 1},{0, 1, 4, 3, 4, 1},101] (* _Ray Chandler_, Aug 26 2015 *)
%t A070431 PowerMod[Range[0,120],2,6] (* or *) PadRight[{},120,{0,1,4,3,4,1}] (* _Harvey P. Dale_, Aug 11 2019 *)
%o A070431 (Sage) [power_mod(n,2,6) for n in range(0, 101)] # _Zerinvary Lajos_, Oct 30 2009
%o A070431 (PARI) a(n)=n^2%6 \\ _Charles R Greathouse IV_, Sep 24 2015
%o A070431 (Magma) [n^2 mod 6 : n in [0..100]]; // _Wesley Ivan Hurt_, Apr 01 2016
%o A070431 (Magma) [Modexp(n, 2, 6): n in [0..100]]; // _Vincenzo Librandi_, Apr 02 2016
%Y A070431 Cf. A000290, A008959, A070435, A070438, A070442, A070452, A159852.
%K A070431 nonn,easy
%O A070431 0,3
%A A070431 _N. J. A. Sloane_, May 12 2002