A183919 Characteristic sequence for sin(2*Pi/n) being rational.

1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset

1

Comments

The sequence is 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, followed by zeros.

In the I. Niven reference a formula for the algebraic degree of 2*sin(2*Pi/n) is found in theorem 3.9. This theorem is attributed to D. H. Lehmer, but the sine part in the Lehmer reference is wrong (to wit: n = 12 has rational value 2*sin(2*Pi/12) = 2*sin(Pi/6) = 1). Hence the degree is 1 = phi(12)/4, as in Niven's book, but not phi(12)/2 = 2 as in Lehmer's paper (the Sines-table there is wrong).

References

I. Niven, Irrational Numbers, The Math. Assoc. of America, second printing, 1963, distributed by John Wiley and Sons.

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000

D. H. Lehmer, A Note on Trigonometric Algebraic Numbers, Am. Math. Monthly 40 (3) (1933) 165-6.

Index entries for characteristic functions

Formula

a(n) = 1 if sin(2*Pi/n) is rational, and a(n) = 0 if it is irrational.

Example

The rational values of 2*sin(2*Pi/n) are 0, 0, 2 and 1 for n=1, 2, 4 and 12, respectively. Otherwise irrational values appear.

Prog

(PARI) A183919(n) = if(n<1, 0, polcoeff( x^1+x^2+x^4+x^12, n)); \\ Antti Karttunen, Dec 24 2018, after code in A089011

Crossrefs

Cf. sequence for cos(2*Pi/n) is A183918.

Sequence in context: A039963 A267537 A329670 * A355449 A058840 A266155

Adjacent sequences: A183916 A183917 A183918 * A183920 A183921 A183922

Keyword

nonn,easy

Author

Wolfdieter Lang, Jan 13 2011

Status

approved