0,2

May be computed found using convergents to the continued fraction for Pi. If cos(a(n)) is near -1, then a(n) is near an odd multiple of Pi. That is, a(n)/(2k+1) is a good rational approximation to Pi with an odd denominator (and continued fractions give good rational approximations).

If a convergent of the continued fraction for Pi has an odd denominator then the corresponding numerator is a term in this sequence. Otherwise add one to the last term in the convergent to get an approximation of Pi with an odd denominator. In this case, we may get a duplicate of the next convergent which we may just ignore.

To illustrate: [3] = 3/1 -> 3; [3,7] = 22/7 -> 22; [3,7,15] = 333/106; 106 is even -> [3,7,16] = 355/113 -> 355; [3,7,15,1] = 355/113 -> 355 (ignore); [3,7,15,1,292] = 103993/33102 -> [3,7,15,1,293] = 104348/33215 -> 104348

z={}; current=1; Timing[ Do[ If[ Cos[ n ]<current, AppendTo[ z, current=Cos[ n ] ] ], {n, 105000} ] ]; z

Cf. A001203 for the continued fraction for Pi.

Sequence in context: A114996 A153256 A137077 this_sequence A119679 A130846 A114101

Adjacent sequences: A046962 A046963 A046964 this_sequence A046966 A046967 A046968

nonn

wouter.meeussen(AT)pandora.be

More terms from Michel ten Voorde (seqfan(AT)tenvoorde.org)

More terms and comments from Jonathan Cross (jcross(AT)wcox.com), Oct 16, 2001