A343055 Decimal expansion of the imaginary part of i^(1/16), or sin(Pi/32).

0, 9, 8, 0, 1, 7, 1, 4, 0, 3, 2, 9, 5, 6, 0, 6, 0, 1, 9, 9, 4, 1, 9, 5, 5, 6, 3, 8, 8, 8, 6, 4, 1, 8, 4, 5, 8, 6, 1, 1, 3, 6, 6, 7, 3, 1, 6, 7, 5, 0, 0, 5, 6, 7, 2, 5, 7, 2, 6, 4, 9, 7, 9, 8, 0, 9, 3, 8, 7, 3, 0, 2, 7, 8, 9, 0, 8, 7, 5, 3, 6, 8, 0, 7, 1, 1, 1, 0, 7, 7, 1, 4, 6, 3, 1, 8, 5, 5, 9, 5, 5, 4, 0, 7, 4, 2, 0, 6, 5, 2, 6, 4, 4, 4, 1

Offset: 0

Author

Seiichi Manyama, Apr 04 2021

Keywords

• nonn
• cons
• easy
• changed

Comments

An algebraic number of degree 16 and denominator 2. - Charles R Greathouse IV, Jan 09 2022

Examples

			0.09801714032956060199419...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..10000
• Michael Penn, The exact value of sin 5⅝°??, YouTube video, 2021.
• Wikipedia, Trigonometric constants expressed in real radicals
• Index entries for algebraic numbers, degree 16

Crossrefs

sin(Pi/m): A010527 (m=3), A010503 (m=4), A019845 (m=5), A323601 (m=7), A182168 (m=8), A019829 (m=9), A019827 (m=10), A019824 (m=12), A232736 (m=14), A019821 (m=15), A232738 (m=16), A241243 (m=17), A019819 (m=18), A019818 (m=20), A343054 (m=24), A019815 (m=30), this sequence (m=32), A019814 (m=36).

Programs

Mathematica

RealDigits[Sin[Pi/32], 10, 100, -1][[1]] (* Amiram Eldar, Apr 27 2021 *)

PARI

imag(I^(1/16))

PARI

sin(Pi/32)

PARI

sqrt(2-sqrt(2+sqrt(2+sqrt(2))))/2

Sage

numerical_approx(sin(pi/32), digits=123) # G. C. Greubel, Sep 30 2022

Formula

Equals (1/2) * sqrt(2-sqrt(2+sqrt(2+sqrt(2)))).

One of the 16 real roots of -128*x^2 +2688*x^4 -21504*x^6 +84480*x^8 +32768*x^16 -131072*x^14 +212992*x^12 -180224*x^10 +1 =0. - R. J. Mathar, Aug 29 2025