

A121506


Minimal polygon values appearing in a certain polygon problem leading to an approximation of Pi.


0



3, 5, 6, 8, 9, 11, 12, 14, 15, 17, 18, 20, 21, 22, 24, 25, 27, 28, 30, 31, 32, 34, 35, 37, 38, 39, 41, 42, 44, 45, 47, 48, 49, 51, 52, 54, 55, 56, 58, 59, 61, 62, 64, 65, 66, 66, 66, 66, 66, 66, 66, 66, 66, 66, 66, 66
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OFFSET

3,1


COMMENTS

Analog of A121500 with n and m roles interchanged.
For a regular mgon circumscribed around a unit circle (area Pi) the arithmetic mean of the areas of this mgon with a regular inscribed ngon is nearest to Pi for n=a(m).
This exercise was inspired by K. R. Popper's remark on sqrt(2)+sqrt(3) which approximates Pi with a 0.15% relative error. See the Popper reference under A121503.


LINKS

Table of n, a(n) for n=3..58.


FORMULA

a(m)=min(abs(F(n,m)),n=3..infinity), m>=3 (checked for n=3..3+500), with F(nm):= ((Fin(n)+Fout(m))/2Pi)/Pi), where Fin(n):=(n/2)*sin(2*Pi/n) and Fout(m):= m*tan(Pi/m). Fin(n) is the area of the regular ngon inscribed in the unit circle. Fout(n) is the area of an regular ngon circumscribing the unit circle. E(n,m) = (F(n,m)pi)/pi is the relative error.


EXAMPLE

m=15, a(15)=21=n: (Fin(21)+Fout(15))/2 = 3.14163887818241 (maple10, 15 digits) leads to a relative error E(21,15)= 0.0000147(rounded).
m=7, a(7)=9=n: F(9,7) leads to error E(9,7)= 0.003122 (rounded).
This is larger than E(8,6), therefore the m value 7 does not appear in A121502.
m=6, a(6)=8=n: (Fin(8)+Fout(6))/2 = sqrt(2) + sqrt(3) has relative error E(8,6)= 0.001487 (rounded). All other inscribed ngons with circumscribed hexagon lead to a larger relative error.


CROSSREFS

Cf. A121502 (values for m for which relative errors E(n, m) decrease).
Sequence in context: A095117 A184675 A089585 * A114119 A186324 A101358
Adjacent sequences: A121503 A121504 A121505 * A121507 A121508 A121509


KEYWORD

nonn,easy


AUTHOR

Wolfdieter Lang, Aug 16 2006


STATUS

approved



