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A046965
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Cos(a(n)) decreases monotonically to -1.
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7
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1, 2, 3, 22, 355, 104348, 208341, 521030, 833719, 1146408, 5419351, 85563208, 165707065, 245850922, 657408909, 1068966896, 3618458675, 6167950454, 21053343141, 1804419559672, 3587785776203, 5371151992734, 14330089761671, 130796280757852
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OFFSET
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1,2
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COMMENTS
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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
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LINKS
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MATHEMATICA
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z={}; current=1; Timing[ Do[ If[ Cos[ n ]<current, AppendTo[ z, current=Cos[ n ] ] ], {n, 105000} ] ]; z
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CROSSREFS
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Cf. A001203 for the continued fraction for Pi.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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Terms a(13) and beyond and comments from Jonathan Cross (jcross(AT)wcox.com), Oct 16 2001
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STATUS
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approved
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