A002665 Continued fraction expansion of Lehmer's constant. (Formerly M1549 N0605)

0, 1, 1, 2, 5, 34, 985, 1151138, 1116929202845, 1480063770341062927127746, 1846425204836010506550936273411258268076151412465

Offset

0,4

References

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Harry J. Smith, Table of n, a(n) for n = 0..12

D. H. Lehmer, A cotangent analogue of continued fractions, Duke Math. J., 4 (1935), 323-340.

D. H. Lehmer, A cotangent analogue of continued fractions, Duke Math. J., 4 (1935), 323-340. [Annotated scanned copy]

G. Xiao, Contfrac

Index entries for continued fractions for constants

Formula

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, ...

Example

0.592632718201636... = 0 + 1/(1 + 1/(1 + 1/(2 + 1/(5 + ...)))). - Harry J. Smith, May 14 2009

Mathematica

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 *)

Prog

(PARI) default(realprecision, 2000); b=0.;
Lehmers=1/tan(suminf(k=1, b=b^2+b+1; (-1)^k*atan(1/b))+Pi/2);
x=contfrac(Lehmers);
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

Crossrefs

Cf. A030125 (decimal expansion).

Cf. A002794, A002795, A002665, A002065.

Starting with n=2, a(n)/a(n-2) are in A096407.

Sequence in context: A206830 A277436 A358688 * A192222 A326946 A241586

Adjacent sequences: A002662 A002663 A002664 * A002666 A002667 A002668

Keyword

nonn,cofr,nice,easy

Author

N. J. A. Sloane

Extensions

More terms from Jeffrey Shallit

First two terms inserted by Harry J. Smith, May 14 2009

Status

approved