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

1,2

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.

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]

Mathematica

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

Prog

(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

Crossrefs

Cf. A172265 (partial sums).

Sequence in context: A293324 A284904 A084215 * A049938 A002460 A266613

Adjacent sequences: A024807 A024808 A024809 * A024811 A024812 A024813

Keyword

nonn

Author

Clark Kimberling

Extensions

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

Status

approved