A022844 a(n) = floor(n*Pi).

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

Offset: 0

Author

Clark Kimberling

Keywords

• nonn

Comments

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

Examples

			a(7)=21 because 7*Pi=21.9911... and a(8)=25 because 8*Pi=25.1327.... a(100000)=314159 because Pi=3.141592...
		

Links

• 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

Crossrefs

Cf. A000796, A054386, A038130, A108591, A127451, A140758.

First differences give A063438.

Programs

Magma

R:=RieldField(10); [Floor(n*Pi(R)): n in [0..80]]; // G. C. Greubel, Sep 28 2018

Maple

a:=n->floor(n*Pi): seq(a(n),n=0..70); # Muniru A Asiru, Sep 28 2018

Mathematica

Floor[Pi Range[0,200]] (* Harvey P. Dale, Aug 27 2024 *)

PARI

vector(80, n, n--; floor(n*Pi)) \\ G. C. Greubel, Sep 28 2018

Formula

a(n)/n converges to Pi because |a(n)/n - Pi| = |a(n) - n*Pi|/n < 1/n. - Hieronymus Fischer, Jan 22 2006

Extensions

Previous Mathematica program replaced by Harvey P. Dale, Aug 27 2024