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%I #32 Sep 08 2022 08:45:05
%S 0,1,8,3,4,5,0,7,8,9,4,11,0,1,8,3,4,5,0,7,8,9,4,11,0,1,8,3,4,5,0,7,8,
%T 9,4,11,0,1,8,3,4,5,0,7,8,9,4,11,0,1,8,3,4,5,0,7,8,9,4,11,0,1,8,3,4,5,
%U 0,7,8,9,4,11,0,1,8,3,4,5,0,7,8,9,4,11,0,1,8,3,4,5,0,7,8,9,4,11,0,1,8,3
%N a(n) = n^3 mod 12, n^5 mod 12.
%C n^5 - n^3 == 0 (mod 12) is shown explicitly for n = 0 to 11, then the induction n -> n+12 for the 5th-order polynomial followed by binomial expansion of (n+12)^k concludes that the zero (mod 12) is periodically extended to the other integers. - _R. J. Mathar_, Jul 23 2009
%H G. C. Greubel, <a href="/A070474/b070474.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1).
%F From _R. J. Mathar_, Jul 23 2009: (Start)
%F a(n) = a(n-12).
%F G.f.: -x*(1 + 8*x + 3*x^2 + 4*x^3 + 5*x^4 + 7*x^6 + 8*x^7 + 9*x^8 + 4*x^9 + 11*x^10)/ ((x-1) *(1+x+x ^2) *(1+x) *(1-x+x^2) *(1+x^2) *(x^4-x^2+1)). (End)
%t Table[Mod[n^3,12],{n,0,200}] (* _Vladimir Joseph Stephan Orlovsky_, Apr 23 2011 *)
%t PowerMod[Range[0,100],3,12] (* _Harvey P. Dale_, Oct 29 2014 *)
%o (Sage) [power_mod(n,7,12)for n in range(0, 100)] # _Zerinvary Lajos_, Oct 28 2009
%o (Magma) [Modexp(n, 3, 12 ): n in [0..100]]; // _Vincenzo Librandi_, Mar 27 2016
%o (PARI) a(n)=n^3%12 \\ _Charles R Greathouse IV_, Apr 06 2016
%Y Cf. A167176.
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_, May 12 2002
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