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

%I A248355 #12 Oct 09 2014 06:55:22
%S A248355 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,
%T A248355 17,17,17,18,18,18,19,19,19,20,20,20,20,21,21,21,21,22,22,22,22,23,23,
%U A248355 23,23,23,24,24,24,24,25,25,25,25,25,26,26,26,26,26,27
%N A248355 Least k such that Pi - k*sin(Pi/k) < 1/(2n).
%C A248355 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.)
%H A248355 Clark Kimberling, <a href="/A248355/b248355.txt">Table of n, a(n) for n = 1..10000</a>
%F A248355 a(n) ~ Pi*sqrt(Pi*n/3). - _Vaclav Kotesovec_, Oct 09 2014
%e A248355 Approximations are shown here:
%e A248355 n    Pi - arch(n)      1/(2n)
%e A248355 1     3.14159...        0.5
%e A248355 2     1.14159...        0.25
%e A248355 3     0.543516...       0.16667
%e A248355 4     0.313166...       0.125
%e A248355 5     0.202666...       0.1
%e A248355 6     0.141593...       0.08333
%e A248355 7     0.105506...       0.07143
%e A248355 8     0.0801252...      0.0625
%e A248355 a(5) = 8 because Pi - arch(8) < 1/10 < Pi - arch(7).
%t A248355 z = 200; p[k_] := p[k] = k*Sin[Pi/k]; N[Table[Pi - p[n], {n, 1, z/10}]]
%t A248355 f[n_] := f[n] = Select[Range[z], Pi - p[#] < 1/(2 n) &, 1]
%t A248355 u = Flatten[Table[f[n], {n, 1, z}]]        (* A248355 *)
%t A248355 v = Flatten[Position[Differences[u], 0]]   (* A248356 *)
%t A248355 w = Flatten[Position[Differences[u], 1]]   (* A248357 *)
%t A248355 f = Table[Floor[1/(Pi - p[n])], {n, 1, z}] (* A248358 *)
%Y A248355 Cf. A248356, A248357, A248358, A248347, A248578.
%K A248355 nonn,easy
%O A248355 1,1
%A A248355 _Clark Kimberling_, Oct 05 2014