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A065802
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How small is the squeezed n-gon? Let s0 be the side of a regular n-gon and s1 the side of the maximal n-gon which can be squeezed between the former and its circumcircle. The n-th entry in the sequence is floor(s0/s1).
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0
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3, 5, 9, 13, 19, 24, 32, 38, 48, 56, 67, 77, 90, 102, 116, 129, 145, 160, 178, 194, 213, 231, 252, 272, 294, 316, 340, 363, 388, 413, 440, 466, 495, 523, 554, 583, 615, 646, 680, 713, 748, 782, 820, 855, 894, 932, 972, 1011, 1053, 1094, 1137, 1180, 1225
(list; graph; refs; listen; history; internal format)
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OFFSET
| 3,1
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COMMENTS
| Closely related to K(n) = (2*n/Pi)*sin(Pi/n)/(1-cos(Pi/n)) as derived from the n-gon with same circumference as the circel squeezed between the large n-gon and its circumcircle.
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REFERENCES
| Bill Taylor, "Little Geometry problem", Newsgroup sci.math, 31-Oct-2001
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FORMULA
| For n=odd: a(n) = floor((1+cos(Pi/n))/(1-cos(Pi/n))) For n=even: a(n) = floor( 2*(2/(tan(Pi/n))^2) + 1 )
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EXAMPLE
| a(3) = 3 as can be seen in Christmas stars: cos(Pi/3)=1/2, thus a(3) = floor((3/2)/(1/2)) = 3. a(4) = 5 as proposed by Bill Taylor in sci.math: tan(Pi/4)=1, thus a(4) = floor(2*(2/1^2) + 1) = 5.
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MATHEMATICA
| f[n_] := If[ OddQ[n], Floor[(1 + Cos[Pi/n]) / (1 - Cos[Pi/n])], Floor[4/(Tan[Pi/n])^2 + 1] ]; Table[ f[n], {n, 3, 60} ]
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CROSSREFS
| Sequence in context: A074133 A078735 A004132 * A118028 A099392 A080827
Adjacent sequences: A065799 A065800 A065801 * A065803 A065804 A065805
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KEYWORD
| easy,nice,nonn
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AUTHOR
| Rainer Rosenthal (r.rosenthal(AT)web.de), Dec 05 2001
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EXTENSIONS
| More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Dec 06 2001
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