A272534 Decimal expansion of the edge length of a regular 15-gon with unit circumradius.

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

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 Viète. 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: A242131 A177266 A356124 * A332522 A173386 A011443

Adjacent sequences: A272531 A272532 A272533 * A272535 A272536 A272537

Keyword

nonn,cons,easy

Author

Stanislav Sykora, May 02 2016

Status

approved