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A022844
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a(n) = floor(n*Pi).
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35
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0, 3, 6, 9, 12, 15, 18, 21, 25, 28, 31, 34, 37, 40, 43, 47, 50, 53, 56, 59, 62, 65, 69, 72, 75, 78, 81, 84, 87, 91, 94, 97, 100, 103, 106, 109, 113, 116, 119, 122, 125, 128, 131, 135, 138, 141, 144, 147, 150, 153, 157, 160, 163, 166, 169, 172, 175, 179, 182, 185, 188, 191, 194
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OFFSET
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0,2
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COMMENTS
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Beatty sequence for Pi.
Differs from A127451 first at a(57). - L. Edson Jeffery, Dec 01 2013
These are the nonnegative integers m satisfying sin(m)*sin(m+1) <= 0. In general, the Beatty sequence of an irrational number r > 1 consists of the numbers m satisfying sin(m*x)*sin((m+1)*x) <= 0, where x = Pi/r. Thus the numbers m satisfying sin(m*x)*sin((m+1)*x) > 0 form the Beatty sequence of r/(1-r). - Clark Kimberling, Aug 21 2014
This can also be stated in terms of the tangent function. These are the nonnegative integers m such that tan(m/2)*tan(m/2+1/2) <= 0. In general, the Beatty sequence of an irrational number r > 1 consists of the numbers m satisfying tan(m*x/2)*tan((m+1)*x/2) <= 0, where x = Pi/r. Thus the numbers m satisfying tan(m*x/2)*tan((m+1)*x/2) > 0 form the Beatty sequence of r/(1-r). - Clark Kimberling, Aug 22 2014
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LINKS
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Ivan Panchenko, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Beatty Sequence.
Index entries for sequences related to Beatty sequences
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FORMULA
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a(n)/n converges to Pi because |a(n)/n - Pi| = |a(n) - n*Pi|/n < 1/n. - Hieronymus Fischer, Jan 22 2006
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EXAMPLE
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a(7)=21 because 7*Pi=21.9911... and a(8)=25 because 8*Pi=25.1327.... a(100000)=314159 because Pi=3.141592...
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MAPLE
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a:=n->floor(n*Pi): seq(a(n), n=0..70); # Muniru A Asiru, Sep 28 2018
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MATHEMATICA
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a[n_]:=Floor[Pi*n]; (* Vladimir Joseph Stephan Orlovsky, Dec 12 2008 *)
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PROG
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(PARI) vector(80, n, n--; floor(n*Pi)) \\ G. C. Greubel, Sep 28 2018
(MAGMA) R:=RieldField(10); [Floor(n*Pi(R)): n in [0..80]]; // G. C. Greubel, Sep 28 2018
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CROSSREFS
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Cf. A000796, A054386, A038130, A108591, A127451, A140758.
First differences give A063438.
Sequence in context: A323422 A071073 A127451 * A260702 A262712 A195934
Adjacent sequences: A022841 A022842 A022843 * A022845 A022846 A022847
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KEYWORD
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nonn
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AUTHOR
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Clark Kimberling
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STATUS
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approved
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