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%I #22 Sep 08 2022 08:44:44
%S 0,3,1,10,1,18,1,26,1,34,1,42,1,50,1,58,1,66,1,74,1,82,1,90,1,98,1,
%T 106,1,114,1,122,1,130,1,138,1,146,1,154,1,162,1,170,1,178,1,186,1,
%U 194,1,202,1,210,1,218,1,226,1,234,1,242,1,250,1,258,1,266,1,274,1,282,1,290,1,298,1
%N Continued fraction for tan(1/4).
%H Harry J. Smith, <a href="/A019427/b019427.txt">Table of n, a(n) for n = 0..20000</a>
%H G. Xiao, <a href="http://wims.unice.fr/~wims/en_tool~number~contfrac.en.html">Contfrac</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,0,-1).
%F From _Colin Barker_, Sep 08 2013: (Start)
%F a(n) = (-1+3*(-1)^n-4*(-1+(-1)^n)*n)/2 for n>1.
%F a(n) = 2*a(n-2)-a(n-4) for n>5.
%F G.f.: x*(x^4-x^3+4*x^2+x+3) / ((x-1)^2*(x+1)^2). (End)
%e 0.25534192122103626650448223... = 0 + 1/(3 + 1/(1 + 1/(10 + 1/(1 + ...)))). - _Harry J. Smith_, Jun 13 2009
%t Join[{0, 3}, LinearRecurrence[{0, 2, 0, -1}, {1, 10, 1, 18}, 100]] (* _Vincenzo Librandi_, Jan 03 2016 *)
%o (PARI) { allocatemem(932245000); default(realprecision, 91000); x=contfrac(tan(1/4)); for (n=0, 20000, write("b019427.txt", n, " ", x[n+1])); } \\ _Harry J. Smith_, Jun 13 2009
%o (PARI) Vec(x*(x^4-x^3+4*x^2+x+3)/((x-1)^2*(x+1)^2) + O(x^100)) \\ _Colin Barker_, Sep 08 2013
%o (Magma) [0,3] cat [(-1+3*(-1)^n-4*(-1+(-1)^n)*n)/2: n in [2..80]]; // _Vincenzo Librandi_, Jan 03 2016
%Y Cf. A161013 (decimal expansion). - _Harry J. Smith_, Jun 13 2009
%K nonn,cofr,easy
%O 0,2
%A _David W. Wilson_
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