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A351582 Decimal expansion of the root of cot(Pi/(s+1)) - csc(Pi/s). 1
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, 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, 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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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...
Diagrams demonstrating this property of regular s-gons are interesting (see links).
LINKS
Luxor, Diagram of Concentric Unit-sided Polygons (s=3 to s=20). See diagrams near the start and near the end of the article. The triangle, square and pentagon intersect.
FORMULA
For integer values of s >= 3:
c(s) = circumcircle radius of unit-sided regular s-gon = csc(Pi/s) / 2,
i(s) = incircle radius of unit-sided regular s-gon = cot(Pi/s) / 2,
d(s) = i(s+1) - c(s),
d(s) <= 0 for s <= 4, d(s) > 0 for s >= 5.
For real values of s:
d(1) = -infinity,
d'(s) > 0 for s > 1,
d(s) = 0 for s = 4.4954747887528...
EXAMPLE
4.4954747887528...
MAPLE
Digits:= 120:
fsolve(cot(Pi/(s+1))-csc(Pi/s), s); # Alois P. Heinz, Feb 16 2022
MATHEMATICA
RealDigits[s /. FindRoot[Cot[Pi/(s + 1)] == Csc[Pi/s], {s, 4}, WorkingPrecision -> 120]][[1]] (* Amiram Eldar, Feb 14 2022 *)
PROG
(PARI) solve(s=4, 5, cotan(Pi/(s+1)) - 1/sin(Pi/s)) \\ Michel Marcus, Feb 14 2022
CROSSREFS
Sequence in context: A160900 A035116 A088613 * A049723 A010661 A051668
KEYWORD
cons,nonn
AUTHOR
Robert B Fowler, Feb 14 2022
STATUS
approved

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Last modified February 24 17:05 EST 2024. Contains 370307 sequences. (Running on oeis4.)