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

Author

Wouter Meeussen

Keywords

• nonn

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

Crossrefs

Cf. A001203 for the continued fraction for Pi.

Programs

Mathematica

z={}; current=1; Timing[ Do[ If[ Cos[ n ]
				

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