A248358 Floor(1/(Pi - n*sin(Pi/n))).

0, 0, 1, 3, 4, 7, 9, 12, 15, 19, 23, 27, 32, 38, 43, 49, 56, 62, 69, 77, 85, 93, 102, 111, 121, 130, 141, 151, 162, 174, 186, 198, 210, 223, 237, 250, 265, 279, 294, 309, 325, 341, 357, 374, 391, 409, 427, 445, 464, 483, 503, 523, 543, 564, 585, 606, 628

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. A248358 and A248355 provide insight into the manner of convergence of arch(n) to Pi. (For the closely related function Arch, see A248347.)

See A248578 for the similar sequence round(1/(Pi-n*sin(Pi/n))). - M. F. Hasler, Oct 08 2014

Examples

			n    Pi - arch(n)    1/(Pi - arch(n))
1     3.14159...       0.3183...
2     1.14159...       0.8759...
3     0.54351...       1.8398...
4     0.31316...       3.1932...
5     0.20266...       4.9342...
6     0.14159...       7.0625...
		

Links

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

Crossrefs

Cf. A248355, A248356, A248357, 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 *)

PARI

a(n)=1\(Pi-n*sin(Pi/n)) \\ M. F. Hasler, Oct 08 2014

Formula

a(n) ~ 6*n^2/Pi^3. - Vaclav Kotesovec, Oct 09 2014