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A161011 Decimal expansion of tan(1/2).

%I A161011 #22 Nov 21 2019 04:10:05
%S A161011 5,4,6,3,0,2,4,8,9,8,4,3,7,9,0,5,1,3,2,5,5,1,7,9,4,6,5,7,8,0,2,8,5,3,
%T A161011 8,3,2,9,7,5,5,1,7,2,0,1,7,9,7,9,1,2,4,6,1,6,4,0,9,1,3,8,5,9,3,2,9,0,
%U A161011 7,5,1,0,5,1,8,0,2,5,8,1,5,7,1,5,1,8,0,6,4,8,2,7,0,6,5,6,2,1,8,5,8,9,1,0,4
%N A161011 Decimal expansion of tan(1/2).
%C A161011 By the Lindemann-Weierstrass theorem, this constant is transcendental. - _Charles R Greathouse IV_, May 13 2019
%H A161011 Harry J. Smith, <a href="/A161011/b161011.txt">Table of n, a(n) for n = 0..20000</a>
%H A161011 MathOverflow, <a href="https://mathoverflow.net/questions/128676/what-is-the-effect-of-adding-1-2-to-a-continued-fraction">What is the effect of adding 1/2 to a continued fraction?</a>
%H A161011 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lindemann%E2%80%93Weierstrass_theorem">Lindemann-Weierstrass theorem</a>
%H A161011 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F A161011 From _Peter Bala_, Nov 17 2019: (Start)
%F A161011 Related simple continued fraction expansions:
%F A161011 tan(1/2) = [0; 1, 1, 4, 1, 8, 1, 12, 1, 16, 1, 20, 1, ...]. See A019425.
%F A161011 2*tan(1/2) = [1, 10, 1, 3, 1, 26, 1, 7, 1, 42, 1, 11, 1, 58, 1, 15, 1, 74, 1, 19, 1, 90, ...]
%F A161011 (1/2)*tan(1/2) = [0; 3, 1, 1, 1, 18, 1, 5, 1, 34, 1, 9, 1, 50, 1, 13, 1, 66, 1, 17, 1, 82, ...].
%F A161011 tan(1/2)/(1 - tan(1/2)) = [1, 4, 1, 8, 1, 12, 1, 16, 1, 20, 1, 24, ...]
%F A161011 2*tan(1/2)/(1 - tan(1/2)) = [2, 2, 2, 4, 2, 6, 2, 8, 2, 10, 2, 12, ...]
%F A161011 4*tan(1/2)/(1 - tan(1/2)) = [4, 1, 4, 2, 4, 3, 4, 4, 4, 5, 4, 6, 4, 7, ...]. (End)
%e A161011 0.546302489843790513255179465780285383297551720179791246164091385932907...
%t A161011 RealDigits[N[Tan[1/2],6! ]][[1]] (* _Vladimir Joseph Stephan Orlovsky_, Jun 13 2009 *)
%o A161011 (PARI) default(realprecision, 20080); x=10*tan(1/2); for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b161011.txt", n, " ", d));
%Y A161011 Cf. A019425 (continued fraction). Cf. A049471, A161011 through A161019.
%K A161011 cons,nonn
%O A161011 0,1
%A A161011 _Harry J. Smith_, Jun 13 2009