A046965 Cos(a(n)) decreases monotonically to -1.

1, 2, 3, 22, 355, 104348, 208341, 521030, 833719, 1146408, 5419351, 85563208, 165707065, 245850922, 657408909, 1068966896, 3618458675, 6167950454, 21053343141, 1804419559672, 3587785776203, 5371151992734, 14330089761671, 130796280757852

Offset

1,2

Comments

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

Links

Table of n, a(n) for n=1..24.

Mathematica

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

Crossrefs

Cf. A001203 for the continued fraction for Pi.

Sequence in context: A114996 A153256 A137077 * A119679 A361669 A191648

Adjacent sequences: A046962 A046963 A046964 * A046966 A046967 A046968

Keyword

nonn

Author

Wouter Meeussen

Extensions

More terms from Michel ten Voorde

Terms a(13) and beyond and comments from Jonathan Cross (jcross(AT)wcox.com), Oct 16 2001

Offset changed to 1 by Alois P. Heinz, Apr 12 2019

Status

approved