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A248347
a(n) = floor(1/(Pi - 2^(n+1)*sin(Pi/2^(n+1)))).
5
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
1,1
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.)
LINKS
FORMULA
a(n) ~ 6 * 4^(n+1) / Pi^3. - Vaclav Kotesovec, Oct 09 2014
EXAMPLE
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...
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 *)
CROSSREFS
KEYWORD
nonn,easy,changed
AUTHOR
Clark Kimberling, Oct 05 2014
STATUS
approved