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A248355
Least k such that Pi - k*sin(Pi/k) < 1/(2n).
6
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, 17, 17, 17, 18, 18, 18, 19, 19, 19, 20, 20, 20, 20, 21, 21, 21, 21, 22, 22, 22, 22, 23, 23, 23, 23, 23, 24, 24, 24, 24, 25, 25, 25, 25, 25, 26, 26, 26, 26, 26, 27
OFFSET
1,1
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. A248355 provides insight into the manner of convergence of arch(n) to Pi. (For the closely related function Arch, see A248347.)
LINKS
FORMULA
a(n) ~ Pi*sqrt(Pi*n/3). - Vaclav Kotesovec, Oct 09 2014
EXAMPLE
Approximations are shown here:
n Pi - arch(n) 1/(2n)
1 3.14159... 0.5
2 1.14159... 0.25
3 0.543516... 0.16667
4 0.313166... 0.125
5 0.202666... 0.1
6 0.141593... 0.08333
7 0.105506... 0.07143
8 0.0801252... 0.0625
a(5) = 8 because Pi - arch(8) < 1/10 < Pi - arch(7).
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 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Oct 05 2014
STATUS
approved