A024810 a(n) = floor( tan(m*Pi/2) ), where m = 1 - 2^(-n).

1, 2, 5, 10, 20, 40, 81, 162, 325, 651, 1303, 2607, 5215, 10430, 20860, 41721, 83443, 166886, 333772, 667544, 1335088, 2670176, 5340353, 10680707, 21361414, 42722829, 85445659, 170891318, 341782637, 683565275, 1367130551, 2734261102, 5468522204, 10937044409

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Comments

Geometrically, each term of the sequence represents the integer part of the distance between opposite vertices and also edges of even sided polygons, each of which has double the number of sides of the previous, starting with a square of unit length. - Torlach Rush, Feb 21 2014

a(n) is the greatest integer k such that k/2^n < 2/Pi. - Clark Kimberling, Oct 10 2017

Number of roots of sin(1/x) = 0 in interval 1/2^(n+1) < x < 1. - Hugo Pfoertner, Oct 24 2019

Or simply: number of zeros of sin(x) in the range [1, 2^(n+1)]. - M. F. Hasler, Oct 25 2019

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Sanjar M. Abrarov, Rajinder K. Jagpal, Rehan Siddiqui, and Brendan M. Quine, Algorithmic determination of a large integer in the two-term Machin-like formula for pi, arXiv:2107.01027 [math.GM], 2021.
• Hugo Pfoertner, Illustration of initial terms, sin(1/x) plotted on logarithmic x axis.

Crossrefs

Cf. A127266 (mod 2), A172265 (partial sums).

Programs

Mathematica

Table[Floor[Tan[(1 - 2^(-n)) Pi/2]], {n, 1, 40}] (* Vincenzo Librandi, Feb 26 2014 *)

PARI

a(n) = floor(tan((1 - 2^(-n))*Pi/2)) \\ Michel Marcus, Mar 23 2013

PARI

A024810(n)=2^(n+1)\Pi \\ M. F. Hasler, Oct 25 2019

Formula

a(n) = floor( 1 / tan( Pi / 2^(n+1) )). - Michael Somos, Feb 24 2014

a(n) = floor(2^(n+1)/Pi). - Clark Kimberling, Oct 10 2017 [Corrected by Michel Marcus, Oct 25 2019]

From Sanjar Abrarov, Jun 20 2024: (Start)

a(n) = floor(c_n/sqrt(2-c_{n-1})), where c_n=sqrt(2+c_{n-1}) and c_0 = 0.

a(n) = 2*a(n-1)+A127266(n). (End)

Extensions

a(30)-a(33) corrected by Michel Marcus, Mar 23 2013