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A002665 Continued fraction expansion of Lehmer's constant.

%I A002665 M1549 N0605 #37 Jan 21 2016 09:00:45
%S A002665 0,1,1,2,5,34,985,1151138,1116929202845,1480063770341062927127746,
%T A002665 1846425204836010506550936273411258268076151412465
%N A002665 Continued fraction expansion of Lehmer's constant.
%D A002665 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002665 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002665 Harry J. Smith, <a href="/A002665/b002665.txt">Table of n, a(n) for n = 0..12</a>
%H A002665 D. H. Lehmer, <a href="http://projecteuclid.org/euclid.dmj/1077490638">A cotangent analogue of continued fractions</a>, Duke Math. J., 4 (1935), 323-340.
%H A002665 D. H. Lehmer, <a href="/A002065/a002065_1.pdf">A cotangent analogue of continued fractions</a>, Duke Math. J., 4 (1935), 323-340. [Annotated scanned copy]
%H A002665 G. Xiao, <a href="http://wims.unice.fr/~wims/en_tool~number~contfrac.en.html">Contfrac</a>
%H A002665 <a href="/index/Con#confC">Index entries for continued fractions for constants</a>
%F A002665 With a different offset: a(0)=1, a(1)=1, a(n+1)=(b(n)+b(n-1)+1)*a(n-1), n >= 1, b()=A002065, with b(0)=0, b(1)=1, b(2)=3, ...
%e A002665 0.592632718201636... = 0 + 1/(1 + 1/(1 + 1/(2 + 1/(5 + ...)))). - _Harry J. Smith_, May 14 2009
%t A002665 digits = 1200; c[0] = 0; c[n_] := c[n] = c[n-1]^2 + c[n-1] + 1; LC[m_] := LC[m] = Cot[Sum[(-1)^k*ArcCot[c[k]], {k, 0, m}]] // N[#, digits+10]&; LC[10]; LC[m = 20]; While[Abs[LC[m] - LC[m-10]] > 10^-digits, m = m+10]; ContinuedFraction[LC[m]] (* _Jean-François Alcover_, Oct 08 2013 *)
%o A002665 (PARI) default(realprecision, 2000);b=0.;
%o A002665 Lehmers=1/tan(suminf(k=1,b=b^2+b+1;(-1)^k*atan(1/b))+Pi/2);
%o A002665 x=contfrac(Lehmers);
%o A002665 for (n=1, 13, write("b002665.txt", n-1, " ", x[n])) \\ _Harry J. Smith_, May 14 2009; edited by _Charles R Greathouse IV_, Jan 21 2016
%Y A002665 Cf. A030125 (decimal expansion).
%Y A002665 Cf. A002794, A002795, A002665, A002065.
%Y A002665 Starting with n=2, a(n)/a(n-2) are in A096407.
%K A002665 nonn,cofr,nice,easy
%O A002665 0,4
%A A002665 _N. J. A. Sloane_
%E A002665 More terms from _Jeffrey Shallit_
%E A002665 First two terms inserted by _Harry J. Smith_, May 14 2009