A022844 - OEIS

%I #67 Aug 27 2024 16:23:59

%S 0,3,6,9,12,15,18,21,25,28,31,34,37,40,43,47,50,53,56,59,62,65,69,72,

%T 75,78,81,84,87,91,94,97,100,103,106,109,113,116,119,122,125,128,131,

%U 135,138,141,144,147,150,153,157,160,163,166,169,172,175,179,182,185,188,191,194

%N a(n) = floor(n*Pi).

%C Beatty sequence for Pi.

%C Differs from A127451 first at a(57). - _L. Edson Jeffery_, Dec 01 2013

%C 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

%C 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

%H Ivan Panchenko, <a href="/A022844/b022844.txt">Table of n, a(n) for n = 0..10000</a>

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

%H <a href="/index/Be#Beatty">Index entries for sequences related to Beatty sequences</a>

%F a(n)/n converges to Pi because |a(n)/n - Pi| = |a(n) - n*Pi|/n < 1/n. - _Hieronymus Fischer_, Jan 22 2006

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

%p a:=n->floor(n*Pi): seq(a(n),n=0..70); # _Muniru A Asiru_, Sep 28 2018

%t Floor[Pi Range[0,200]] (* _Harvey P. Dale_, Aug 27 2024 *)

%o (PARI) vector(80, n, n--; floor(n*Pi)) \\ _G. C. Greubel_, Sep 28 2018

%o (Magma) R:=RieldField(10); [Floor(n*Pi(R)): n in [0..80]]; // _G. C. Greubel_, Sep 28 2018

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

%Y First differences give A063438.

%K nonn

%O 0,2

%A _Clark Kimberling_

%E Previous Mathematica program replaced by _Harvey P. Dale_, Aug 27 2024