A248355 Least k such that Pi - k*sin(Pi/k) < 1/(2n).

4, 5, 6, 7, 8, 8, 9, 10, 10, 11, 11, 12, 12, 13, 13, 13, 14, 14, 14, 15, 15, 16, 16, 16, 17, 17, 17, 17, 18, 18, 18, 19, 19, 19, 20, 20, 20, 20, 21, 21, 21, 21, 22, 22, 22, 22, 23, 23, 23, 23, 23, 24, 24, 24, 24, 25, 25, 25, 25, 25, 26, 26, 26, 26, 26, 27

Offset: 1

Author

Clark Kimberling, Oct 05 2014

Keywords

• nonn
• easy

Comments

For n > 1, let arch(n) = n*sin(Pi/n) be the Archimedean approximation to Pi (Finch, pp. 17 and 23) given by a regular polygon of n+1 sides. A248355 provides insight into the manner of convergence of arch(n) to Pi. (For the closely related function Arch, see A248347.)

Examples

			Approximations are shown here:
n    Pi - arch(n)      1/(2n)
1     3.14159...        0.5
2     1.14159...        0.25
3     0.543516...       0.16667
4     0.313166...       0.125
5     0.202666...       0.1
6     0.141593...       0.08333
7     0.105506...       0.07143
8     0.0801252...      0.0625
a(5) = 8 because Pi - arch(8) < 1/10 < Pi - arch(7).
		

Links

• Clark Kimberling, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A248356, A248357, A248358, A248347, A248578.

Programs

Mathematica

z = 200; p[k_] := p[k] = k*Sin[Pi/k]; N[Table[Pi - p[n], {n, 1, z/10}]]
f[n_] := f[n] = Select[Range[z], Pi - p[#] < 1/(2 n) &, 1]
u = Flatten[Table[f[n], {n, 1, z}]]        (* A248355 *)
v = Flatten[Position[Differences[u], 0]]   (* A248356 *)
w = Flatten[Position[Differences[u], 1]]   (* A248357 *)
f = Table[Floor[1/(Pi - p[n])], {n, 1, z}] (* A248358 *)

Formula

a(n) ~ Pi*sqrt(Pi*n/3). - Vaclav Kotesovec, Oct 09 2014