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Search: id:A121501
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| A121501 |
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Positions n of A121500 where the minimal relative error associated with the polygon problem described there decreases. |
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3
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| 3, 5, 6, 8, 11, 14, 15, 17, 18, 21, 31, 38, 48, 65, 82, 89, 99, 106, 123, 181, 222, 280, 379, 478, 519, 577, 618, 717
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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The minimal relative errors for the unit circle area approximation by the arithmetic mean of areas of an inscribed regular n-gon and a circumscribed regular A121500(n)-gon decrease (strictly) for these n=a(k) values. This results from a minimization, first within row n and then along the rows n of the matrix E(n,m) defined below.
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LINKS
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W. Lang: Sequence of decreasing relative errors and more.
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FORMULA
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a(k) is such that E(a(k),A121500(a(k)) < min(E(n,A121500(n)),n=3..a(k)-1), k>=2, a(1):=3, with the relative error E(n,m):= abs(F(n,m)-Pi))/Pi and F(n,m):= (Fin(n)+Fout(m))/2, where Fin(n):=(n/2)*sin(2*Pi/ n) and Fout(m):= m*tan(Pi/m).
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EXAMPLE
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k=4, a(4)=8, m:= A121500(8)= 6. The relative error associated with
F(n=8,m=6) is the smallest among those with values n=3,..,8.
(n,m) pairs (a(k),A121500(a(k)),k=1..7: [3, 3], [5, 4], [6, 5], [8,
6], [11, 8], [14, 10], [15, 11],...
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CROSSREFS
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Cf. A121502 (corresponding A121500(a(k)) numbers).
Sequence in context: A154111 A047444 A160734 this_sequence A157017 A062832 A089085
Adjacent sequences: A121498 A121499 A121500 this_sequence A121502 A121503 A121504
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KEYWORD
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nonn,more
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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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