A248347 a(n) = floor(1/(Pi - 2^(n+1)*sin(Pi/2^(n+1)))).

3, 12, 49, 198, 792, 3170, 12681, 50727, 202909, 811636, 3246545, 12986183, 51944732, 207778928, 831115713, 3324462855, 13297851421, 53191405684, 212765622737, 851062490950, 3404249963800, 13616999855201, 54467999420806, 217871997683226, 871487990732903

Offset: 1

Author

Clark Kimberling, Oct 05 2014

Keywords

• nonn
• easy

Comments

Let Arch(n) = 2^(n+1)*sin(Pi/2^(n+1)) be the Archimedean approximation to Pi (Finch, pp. 17 and 23) given by a regular polygon of 2^(n+1) sides. A248347 provides insight into the manner of convergence of Arch(n) to Pi. Another provider is the fact that the least k for which Arch(k) < 1/4^n is A000027(n) = n. (For the closely related function arch, see A248355.)

Examples

			n    Pi - Arch(n)      1/(Pi - Arch(n))
1    0.313166...         3.1932...
2    0.0801252...       12.4805...
3    0.0201475...       49.6339...
4    0.00504416...     198.249...
5    0.0012615...      792.709...
		

Links

• Clark Kimberling, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000027, A248355, A248357, A248355, A248360.

Programs

Mathematica

z = 200; p[k_] := p[k] = (2^(k + 1))*Sin[Pi/2^(k + 1)]
Table[Floor[1/(Pi - p[n])], {n, 1, z}]  (* A248347  *)

Formula

a(n) ~ 6 * 4^(n+1) / Pi^3. - Vaclav Kotesovec, Oct 09 2014