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%I #32 Feb 29 2024 15:57:12
%S 5,8,18,21,31,34,44,47,57,60,70,73,83,86,96,99,109,112,122,125,135,
%T 138,148,151,161,164,174,177,187,190,200,203,213,216,226,229,239,242,
%U 252,255,265,268,278,281,291,294,304,307,317,320,330,333,343,346,356,359
%N Numbers k such that k^2 == -1 (mod 13).
%C Numbers k such that k == 5 or 8 mod 13. - _Charles R Greathouse IV_, Dec 28 2011
%H Vincenzo Librandi, <a href="/A155086/b155086.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1).
%F a(n) = a(n-1)+a(n-2)-a(n-3).
%F G.f.: x*(5+3*x+5*x^2)/((1+x)*(x-1)^2) .
%F a(n) = 13*(n-1/2)/2 -7*(-1)^n/4.
%F a(n) = a(n-2)+13. - _M. F. Hasler_, Jun 16 2010
%F Sum_{n>=1} (-1)^(n+1)/a(n) = tan(3*Pi/26)*Pi/13. - _Amiram Eldar_, Feb 27 2023
%t LinearRecurrence[{1, 1, -1}, {5, 8, 18}, 50] (* _Vincenzo Librandi_, Feb 26 2012 *)
%t Select[Range[1000], PowerMod[#, 2, 13] == 12 &] (* _Vincenzo Librandi_, Apr 24 2014 *)
%o (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 Cf. A002144, A047221 (m=5), A155095 (m=17), A156619 (m=25), A155096 (m=29), A155097 (m=37), A155098 (m=41), A154609 (bisection).
%K nonn,easy
%O 1,1
%A _Vincenzo Librandi_, Jan 20 2009
%E Algebra simplified by _R. J. Mathar_, Aug 18 2009
%E Edited by _N. J. A. Sloane_, Jun 23 2010
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