A343056 Decimal expansion of the real part of i^(1/16), or cos(Pi/32).

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

Offset

0,1

Links

G. C. Greubel, Table of n, a(n) for n = 0..10000

Leon D. Fairbanks, Powers of Cosine and Sine, arXiv:2308.04437 [math.GM], 2023. See p. 3.

Wikipedia, Trigonometric constants expressed in real radicals.

Index entries for algebraic numbers, degree 16

Formula

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

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

Equals 2F1(-1/16,1/16;1/2;1/2). - R. J. Mathar, Aug 31 2025

Example

0.9951847266721968862448369...

Mathematica

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

Prog

(PARI) real(I^(1/16))
(PARI) cos(Pi/32)
(PARI) sqrt(2+sqrt(2+sqrt(2+sqrt(2))))/2
(Magma) R:= RealField(127); Cos(Pi(R)/32); // G. C. Greubel, Sep 30 2022
(SageMath) numerical_approx(cos(pi/32), digits=122) # G. C. Greubel, Sep 30 2022

Crossrefs

cos(Pi/m): A010503 (m=4), A019863 (m=5), A010527 (m=6), A073052 (m=7), A144981 (m=8), A019879 (m=9), A019881 (m=10), A019884 (m=12), A232735 (m=14), A019887 (m=15), A232737 (m=16), A210649 (m=17), A019889 (m=18), A019890 (m=20), A144982 (m=24), A019893 (m=30). this sequence (m=32), A019894 (m=36).

Sequence in context: A392301 A192106 A346572 * A379732 A117232 A155995

Adjacent sequences: A343053 A343054 A343055 * A343057 A343058 A343059

Keyword

nonn,cons,easy

Author

Seiichi Manyama, Apr 04 2021

Status

approved