OFFSET
1,4
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
LINKS
Clark Kimberling, Table of n, a(n) for n = 1..10000
FORMULA
a(n) ~ 6*n^2/Pi^3. - Vaclav Kotesovec, Oct 09 2014
EXAMPLE
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...
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 *)
PROG
(PARI) a(n)=1\(Pi-n*sin(Pi/n)) \\ M. F. Hasler, Oct 08 2014
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Oct 05 2014
STATUS
approved