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

%I A248358 #13 Oct 09 2014 05:53:36
%S A248358 0,0,1,3,4,7,9,12,15,19,23,27,32,38,43,49,56,62,69,77,85,93,102,111,
%T A248358 121,130,141,151,162,174,186,198,210,223,237,250,265,279,294,309,325,
%U A248358 341,357,374,391,409,427,445,464,483,503,523,543,564,585,606,628
%N A248358 Floor(1/(Pi - n*sin(Pi/n))).
%C A248358 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.)
%C A248358 See A248578 for the similar sequence round(1/(Pi-n*sin(Pi/n))). - _M. F. Hasler_, Oct 08 2014
%H A248358 Clark Kimberling, <a href="/A248358/b248358.txt">Table of n, a(n) for n = 1..10000</a>
%F A248358 a(n) ~ 6*n^2/Pi^3. - _Vaclav Kotesovec_, Oct 09 2014
%e A248358 n    Pi - arch(n)    1/(Pi - arch(n))
%e A248358 1     3.14159...       0.3183...
%e A248358 2     1.14159...       0.8759...
%e A248358 3     0.54351...       1.8398...
%e A248358 4     0.31316...       3.1932...
%e A248358 5     0.20266...       4.9342...
%e A248358 6     0.14159...       7.0625...
%t A248358 z = 200; p[k_] := p[k] = k*Sin[Pi/k]; N[Table[Pi - p[n], {n, 1, z/10}]]
%t A248358 f[n_] := f[n] = Select[Range[z], Pi - p[#] < 1/(2 n) &, 1]
%t A248358 u = Flatten[Table[f[n], {n, 1, z}]]        (* A248355 *)
%t A248358 v = Flatten[Position[Differences[u], 0]]   (* A248356 *)
%t A248358 w = Flatten[Position[Differences[u], 1]]   (* A248357 *)
%t A248358 f = Table[Floor[1/(Pi - p[n])], {n, 1, z}] (* A248358 *)
%o A248358 (PARI) a(n)=1\(Pi-n*sin(Pi/n)) \\ _M. F. Hasler_, Oct 08 2014
%Y A248358 Cf. A248355, A248356, A248357, A248347, A248578.
%K A248358 nonn,easy
%O A248358 1,4
%A A248358 _Clark Kimberling_, Oct 05 2014