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%I #30 Mar 04 2022 07:34:06
%S 4,4,9,5,4,7,4,7,8,8,7,5,2,8,8,9,0,1,6,0,7,1,7,2,3,7,9,6,0,2,8,9,3,2,
%T 9,9,3,6,6,9,0,5,1,5,6,1,3,5,4,8,6,0,9,5,6,5,9,8,3,0,5,6,9,5,4,3,8,8,
%U 0,7,3,9,3,3,5,0,3,7,9,2,0,2,6,9,2,4,0,5,4,9,2,6,1,9,5,4,2,5,8,1,9,4,4,3,1,7
%N Decimal expansion of the root of cot(Pi/(s+1)) - csc(Pi/s).
%C For regular unit-sided polygons with number of sides s >= 3, the s-gon fits inside the (s+1)-gon, and hence inside any t-gon where t > s. For s = 3 and s = 4, this is verified by diagram. For s >= 5, it is verified by observing that the s-gon's circumcircle is smaller than the (s+1)-gon's incircle. The difference of the two circles' radii is negative for s <= 4 and positive for s >= 5, and changes sign at non-integer value s = 4.49547...
%C Diagrams demonstrating this property of regular s-gons are interesting (see links).
%H Robert B Fowler, <a href="/A351582/a351582.pdf">Diagram of Nested Unit-sided Regular Polygons (s=3 to s=12)</a>
%H Luxor, <a href="http://juliagraphics.github.io/Luxor.jl/v0.9.1/polygons.html">Diagram of Concentric Unit-sided Polygons (s=3 to s=20)</a>. See diagrams near the start and near the end of the article. The triangle, square and pentagon intersect.
%F For integer values of s >= 3:
%F c(s) = circumcircle radius of unit-sided regular s-gon = csc(Pi/s) / 2,
%F i(s) = incircle radius of unit-sided regular s-gon = cot(Pi/s) / 2,
%F d(s) = i(s+1) - c(s),
%F d(s) <= 0 for s <= 4, d(s) > 0 for s >= 5.
%F For real values of s:
%F d(1) = -infinity,
%F d'(s) > 0 for s > 1,
%F d(s) = 0 for s = 4.4954747887528...
%e 4.4954747887528...
%p Digits:= 120:
%p fsolve(cot(Pi/(s+1))-csc(Pi/s),s); # _Alois P. Heinz_, Feb 16 2022
%t RealDigits[s /. FindRoot[Cot[Pi/(s + 1)] == Csc[Pi/s], {s, 4}, WorkingPrecision -> 120]][[1]] (* _Amiram Eldar_, Feb 14 2022 *)
%o (PARI) solve(s=4, 5, cotan(Pi/(s+1)) - 1/sin(Pi/s)) \\ _Michel Marcus_, Feb 14 2022
%K cons,nonn
%O 1,1
%A _Robert B Fowler_, Feb 14 2022
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