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
Examples
0.9951847266721968862448369...
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
Crossrefs
Programs
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Magma
R:= RealField(127); Cos(Pi(R)/32); // G. C. Greubel, Sep 30 2022
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Mathematica
RealDigits[Cos[Pi/32], 10, 100][[1]] (* Amiram Eldar, Apr 27 2021 *)
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PARI
real(I^(1/16))
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PARI
cos(Pi/32)
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PARI
sqrt(2+sqrt(2+sqrt(2+sqrt(2))))/2
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SageMath
numerical_approx(cos(pi/32), digits=122) # G. C. Greubel, Sep 30 2022
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