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A272534 Decimal expansion of the edge length of a regular 15-gon with unit circumradius. 7
4, 1, 5, 8, 2, 3, 3, 8, 1, 6, 3, 5, 5, 1, 8, 6, 7, 4, 2, 0, 3, 4, 8, 4, 5, 6, 8, 8, 1, 0, 2, 5, 0, 3, 3, 2, 4, 3, 3, 1, 6, 9, 5, 2, 1, 2, 5, 5, 4, 4, 7, 6, 7, 2, 8, 1, 4, 3, 6, 3, 9, 4, 7, 7, 6, 4, 7, 6, 5, 6, 5, 1, 3, 2, 8, 1, 4, 8, 7, 5, 2, 6, 0, 9, 2, 5, 7, 5, 1, 3, 4, 4, 5, 4, 5, 5, 1, 4, 6, 1, 1, 5, 7, 3, 0 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

15-gon is the first m-gon with odd composite m which is constructible (see A003401) in virtue of the fact that 15 is the product of two distinct Fermat primes (A019434). The next such case is 51-gon (m=3*17), followed by 85-gon (m=5*17), 771-gon (m=3*257), etc.

From Wolfdieter Lang, Apr 29 2018: (Start)

This constant appears in a historic problem posed by Adriaan van Roomen (Adrianus Romanus) in his Ideae mathematicae from 1593, solved by Vi├Ęte. See the Havil reference, problem 4, pp. 69-74. See also the comments in A302711 with a link to Romanus' book, Exemplum quaesitum.

This problem is equivalent to R(45, 2*sin(Pi/675)) = 2*sin(Pi/15), with a special case of monic Chebyshev polynomials of the first kind, named R, given in A127672. For the constant 2*sin(Pi/675) see A302716. (End)

REFERENCES

Julian Havil, The Irrationals, A Story of the Numbers You Can't Count On, Princeton University Press, Princeton and Oxford, 2012, pp. 69-74.

LINKS

Stanislav Sykora, Table of n, a(n) for n = 0..2000

Mauro Fiorentini, Construibili (numeri)

Eric Weisstein's World of Mathematics, Constructible Number

Wikipedia, Constructible number

Wikipedia, Regular polygon

Index entries for sequences related to Chebyshev polynomials.

FORMULA

Equals 2*sin(Pi/m) for m=15.

Also equals (sqrt(3) - sqrt(15) + sqrt(10 + 2*sqrt(5)))/4.

Also equals sqrt(7 - sqrt(5) - sqrt(30 - 6*sqrt(5)))/2. This is the rewritten expression of the Havil reference on top of p. 70. - Wolfdieter Lang, Apr 29 2018

EXAMPLE

0.415823381635518674203484568810250332433169521255447672814363947...

MATHEMATICA

RealDigits[N[2Sin[Pi/15], 100]][[1]] (* Robert Price, May 02 2016*)

PROG

(PARI) 2*sin(Pi/15)

CROSSREFS

Cf. A003401, A019434, A127672, A302711, A302716.

Edge lengths of other constructible m-gons: A002194 (m=3), A002193 (4), A182007 (5), A101464 (8), A094214 (10), A101263 (12), A272535 (16), A228787 (17), A272536 (20).

Sequence in context: A133866 A242131 A177266 * A173386 A011443 A016687

Adjacent sequences:  A272531 A272532 A272533 * A272535 A272536 A272537

KEYWORD

nonn,cons,easy

AUTHOR

Stanislav Sykora, May 02 2016

STATUS

approved

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Last modified July 21 21:23 EDT 2018. Contains 312880 sequences. (Running on oeis4.)