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Search: id:A121500
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| A121500 |
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Minimal polygon values for a certain polygon problem leading to an approximation of pi. |
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+0
5
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| 3, 4, 4, 5, 6, 6, 7, 7, 8, 9, 9, 10, 11, 12, 12, 13, 14, 14, 15, 16, 16, 17, 18, 19, 19, 20, 21, 21, 22, 23, 23, 24, 25, 26, 26, 27, 28, 28, 29, 30, 30, 31, 32, 33, 33, 34, 35, 35, 36, 37, 38, 38, 39, 40, 40, 41, 42, 42
(list; graph; listen)
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OFFSET
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3,1
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COMMENT
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For a regular n-gon inscribed in a unit circle (area pi) the arithmetic mean of the areas of this n-gon with a regular circumsribed m-gon is nearest to pi for m=a(n).
This exercise was inspired by K. R. Popper's remark on sqrt(2)+sqrt(3) which approximates Pi with 1.5 permille relative error. See the Popper reference under A121503.
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FORMULA
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a(n)=min(abs(E(n,m)),m=3..infinity), n>=3 (checked for m=3..3+500), with E(n,m):= ((Fin(n)+Fout(m))/2-Pi)/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 n-gon inscribed in the unit circle. Fout(n) is the area of a regular n-gon circumscribing the unit circle.
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EXAMPLE
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n=8, a(8)=6: (Fin(8)+Fout(6))/2 = sqrt(2) + sqrt(3) has relative error 0.001487 (rounded). All other circumscribed m-gons with inscribed 8-gon lead to a larger relative error.
n=21, a(21)=15: (Fin(21)+Fout(15))/2 = 3.14163887818241 (maple10, 15 digits) leads to a relative error 0.0000147 (rounded).
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CROSSREFS
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Cf. A121501 (positions n where relative errors decrease).
Sequence in context: A054760 A079107 A023963 this_sequence A113455 A054637 A120172
Adjacent sequences: A121497 A121498 A121499 this_sequence A121501 A121502 A121503
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KEYWORD
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nonn,easy
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AUTHOR
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Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Aug 16 2006
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