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a(n) = 2^n mod 19.
6

%I #36 Sep 08 2022 08:44:52

%S 1,2,4,8,16,13,7,14,9,18,17,15,11,3,6,12,5,10,1,2,4,8,16,13,7,14,9,18,

%T 17,15,11,3,6,12,5,10,1,2,4,8,16,13,7,14,9,18,17,15,11,3,6,12,5,10,1,

%U 2,4,8,16,13,7,14,9,18,17,15

%N a(n) = 2^n mod 19.

%C The sequence can be generated via a(n) = A061762(a(n-1)). Apparently any other choice of the first element leads also to periodic sequences, with fixed points of A061762 as special cases. - _Zak Seidov_, Aug 22 2007

%D I. M. Vinogradov, Elements of Number Theory, pp. 220 ff.

%H Vincenzo Librandi, <a href="/A036120/b036120.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (1, 0, 0, 0, 0, 0, 0, 0, -1, 1).

%F a(n)= +a(n-1) -a(n-9) +a(n-10). - _R. J. Mathar_, Apr 13 2010

%F G.f.: (1+x+2*x^2+4*x^3+8*x^4-3*x^5-6*x^6+7*x^7-5*x^8+10*x^9)/ ((1-x) * (1+x) * (x^2- x+1) * (x^6-x^3+1)). - _R. J. Mathar_, Apr 13 2010

%F a(n) = a(n+18). - _Vincenzo Librandi_, Sep 09 2011

%p with(numtheory) ; i := pi(19) ; [ seq(primroot(ithprime(i))^j mod ithprime(i), j=0..100) ];

%t PowerMod[2, Range[0, 100], 19] (* _G. C. Greubel_, Oct 17 2018 *)

%o (Sage) [power_mod(2,n,19) for n in range(0,66)] # _Zerinvary Lajos_, Nov 03 2009

%o (PARI) a(n)=lift(Mod(2,19)^n) \\ _Charles R Greathouse IV_, Mar 22 2016

%o (Magma) [Modexp(2, n, 19): n in [0..100]]; // _G. C. Greubel_, Oct 17 2018

%o (Python) for n in range(0, 100): print(int(pow(2, n, 19)), end=' ') # _Stefano Spezia_, Oct 17 2018

%o (GAP) List([0..60],n->PowerMod(2,n,19)); # _Muniru A Asiru_, Oct 17 2018

%Y CF. A000079 (2^n).

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_