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A155086 Numbers k such that k^2 == -1 (mod 13).

%I A155086 #36 Nov 27 2024 07:25:07
%S A155086 5,8,18,21,31,34,44,47,57,60,70,73,83,86,96,99,109,112,122,125,135,
%T A155086 138,148,151,161,164,174,177,187,190,200,203,213,216,226,229,239,242,
%U A155086 252,255,265,268,278,281,291,294,304,307,317,320,330,333,343,346,356,359
%N A155086 Numbers k such that k^2 == -1 (mod 13).
%C A155086 Numbers k such that k == 5 or 8 mod 13. - _Charles R Greathouse IV_, Dec 28 2011
%H A155086 Vincenzo Librandi, <a href="/A155086/b155086.txt">Table of n, a(n) for n = 1..1000</a>
%H A155086 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1).
%F A155086 a(n) = a(n-1)+a(n-2)-a(n-3).
%F A155086 G.f.: x*(5+3*x+5*x^2)/((1+x)*(x-1)^2) .
%F A155086 a(n) = 13*(n-1/2)/2 -7*(-1)^n/4.
%F A155086 a(n) = a(n-2)+13. - _M. F. Hasler_, Jun 16 2010
%F A155086 Sum_{n>=1} (-1)^(n+1)/a(n) = tan(3*Pi/26)*Pi/13. - _Amiram Eldar_, Feb 27 2023
%F A155086 From _Amiram Eldar_, Nov 25 2024: (Start)
%F A155086 Product_{n>=1} (1 - (-1)^n/a(n)) = cos(Pi/26)*sec(3*Pi/26) = 1/(2*cos(Pi/13)-1).
%F A155086 Product_{n>=1} (1 + (-1)^n/a(n)) = cosec(5*Pi/26)/2. (End)
%t A155086 LinearRecurrence[{1, 1, -1}, {5, 8, 18}, 50] (* _Vincenzo Librandi_, Feb 26 2012 *)
%t A155086 Select[Range[1000], PowerMod[#, 2, 13] == 12 &] (* _Vincenzo Librandi_, Apr 24 2014 *)
%o A155086 (Magma) I:=[5, 8, 18]; [n le 3 select I[n] else Self(n-1)+Self(n-2)-Self(n-3): n in [1..40]]; // _Vincenzo Librandi_, Feb 26 2012
%Y A155086 Cf. A002144, A047221 (m=5), A155095 (m=17), A156619 (m=25), A155096 (m=29), A155097 (m=37), A155098 (m=41), A154609 (bisection).
%K A155086 nonn,easy
%O A155086 1,1
%A A155086 _Vincenzo Librandi_, Jan 20 2009
%E A155086 Algebra simplified by _R. J. Mathar_, Aug 18 2009
%E A155086 Edited by _N. J. A. Sloane_, Jun 23 2010