%I #43 May 15 2022 04:30:02

%S 1,3,22,333,355,103993,104348,208341,312689,833719,1146408,4272943,

%T 5419351,80143857,165707065,245850922,411557987,1068966896,2549491779,

%U 6167950454,14885392687,21053343141,1783366216531,3587785776203

%N |sin(n)| (or |tan(n)| or |sec(n)|) decreases monotonically to 0; also |cos(n)| (or |cosec(n)| or |cot(n)|) increases.

%C Also numerators of convergents to Pi (A002486 gives denominators) beginning at 1.

%C Integer circumferences of circles with a(0)=1 and a(n+1) is the smallest integer circumference with corresponding diameter nearer an integer than is the diameter of the circle with circumference a(n). See PARI program. - _Rick L. Shepherd_, Oct 06 2007

%D K. H. Rosen et al., eds., Handbook of Discrete and Combinatorial Mathematics, CRC Press, 2000; p. 293.

%D Suggested by a question from Alan Walker (Alan_Walker(AT)sabre.com)

%H Michael De Vlieger, <a href="/A046947/b046947.txt">Table of n, a(n) for n = 0..1946</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Cosecant.html">Cosecant</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FlintHillsSeries.html">Flint Hills Series</a>

%e |sin(4272943)| = 0.000000549579497810490800503139..., |tan(4272943)| = 0.000000549579497810573797346111..., |sec(4272943)| = 1.00000000000015101881221...

%e |cos(4272943)| = 0.999999999999848981187793172965367089856..., |cosec(4272943)| = 1819572.97167010734684889..., |cot(4272943)| = 1819572.97166983255709999...

%p Digits := 50; M := 10000; a := [ 1 ]; R := sin(1.); for n from 2 to M do t1 := evalf(sin(n)); if abs(t1)<R then R := abs(t1); a := [ op(a), n ]; fi; od: a;

%p with(numtheory): cf := cfrac (Pi,100): seq(nthnumer(cf,i), i=-1..22 ); # _Zerinvary Lajos_, Feb 07 2007

%t z={}; current=1; Do[ If[ Abs[ Sin[ n]] < current, AppendTo[ z, current=Abs[ Sin[ n]]]], {n, 1, 10^7}]; z (* or *)

%t Join[{1}, Table[ Numerator[ FromContinuedFraction[ ContinuedFraction[Pi, n]]], {n, 1, 23}]] (* _Wouter Meeussen_ *)

%t Join[{1},Convergents[Pi,30]//Numerator] (* _Harvey P. Dale_, May 05 2019 *)

%o (PARI) /* Program calculates a(n) without using sin or continued fraction functions */ {d=1/Pi; print1("1, "); for(circum=2,500000000, dm=circum/Pi; dmin=min(dm-floor(dm),ceil(dm)-dm); if(dmin<d, print1(circum,", "); d=dmin))} /* or could use dmin=min(frac(dm),1-frac(dm)) above */ \\ _Rick L. Shepherd_, Oct 06 2007

%Y Cf. A004112, A049946. See also A002485, which is the same sequence but begins at 0.

%K nonn,nice

%O 0,2

%A _N. J. A. Sloane_

%E More terms from _Wouter Meeussen_

%E Further terms from _Michel ten Voorde_

%E Edited and extended by _Robert G. Wilson v_, Jan 28 2003

%E Typo in examples fixed by _Paolo Bonzini_, Mar 21 2012