A002665 Continued fraction expansion of Lehmer's constant.

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• cofr
• nice
• easy

Examples

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

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

Crossrefs

Cf. A030125 (decimal expansion).

Cf. A002794, A002795, A002665, A002065.

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

Programs

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 *)

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

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, ...

Extensions

More terms from Jeffrey Shallit

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