A232738 Decimal expansion of the imaginary part of I^(1/8), or sin(Pi/16).

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

Offset: 0

Author

Stanislav Sykora, Nov 29 2013

Keywords

• nonn
• cons
• easy
• changed

Comments

The corresponding real part is in A232737.

Examples

			0.195090322016128267848284868477022240927691617751954807754502...
		

Links

• Stanislav Sykora, Table of n, a(n) for n = 0..1000
• Wikipedia, Trigonometric constants expressed in real radicals
• Index entries for algebraic numbers, degree 8

Crossrefs

Cf. A232737 (real part), A010503 (imag(I^(1/2))), A182168 (imag(I^(1/4))), A019827 (imag(I^(1/5))), A019824 (imag(I^(1/6))), A232736 (imag(I^(1/7))), A019819 (imag(I^(1/9))), A019818 (imag(I^(1/10))).

Programs

Magma

R:= RealField(116); Sin(Pi(R)/16); // G. C. Greubel, Sep 20 2022

Mathematica

RealDigits[Sin[Pi/16],10,120][[1]] (* Harvey P. Dale, Sep 01 2018 *)

PARI

imag(I^(1/8)) \\ Seiichi Manyama, Apr 04 2021

PARI

sin(Pi/16) \\ Seiichi Manyama, Apr 04 2021

PARI

sqrt(2-sqrt(2+sqrt(2)))/2 \\ Seiichi Manyama, Apr 04 2021

SageMath

numerical_approx(sin(pi/16), digits=116) # G. C. Greubel, Sep 20 2022

Formula

Equals (1/2) * sqrt(2-sqrt(2+sqrt(2))). - Seiichi Manyama, Apr 04 2021

This^2 + A232737^2 = 1.

Smallest positive of the 8 real-valued roots of 128*x^8-256*x^6+160*x^4-32*x^2+1=0.