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A144981 Decimal expansion of cos(Pi/8) = cos(22.5 degrees).

%I A144981 #55 Aug 31 2025 10:09:53
%S A144981 9,2,3,8,7,9,5,3,2,5,1,1,2,8,6,7,5,6,1,2,8,1,8,3,1,8,9,3,9,6,7,8,8,2,
%T A144981 8,6,8,2,2,4,1,6,6,2,5,8,6,3,6,4,2,4,8,6,1,1,5,0,9,7,7,3,1,2,8,0,5,3,
%U A144981 5,0,0,7,5,0,1,1,0,2,3,5,8,7,1,4,8,3,9,9,3,4,8,5,0,3,4,4,5,9,6,0,9,7,9,6,3
%N A144981 Decimal expansion of cos(Pi/8) = cos(22.5 degrees).
%C A144981 Also the real part of i^(1/4). - _Stanislav Sykora_, Apr 25 2012
%C A144981 Width of a regular octagon of unit diameter. See Bingane and Audet. - _Michel Marcus_, Oct 04 2021
%C A144981 Minimal polynomial 8x^4 - 8x^2 + 1. - _Charles R Greathouse IV_, Oct 30 2023
%C A144981 Also the ratio (1+sqrt(2))/sqrt(4+2*sqrt(2)) of the radii and perimeters of the inscribed and circumscribed circles of a regular octagon. This and the first two comments are actually all equivalent. - _M. F. Hasler_, Aug 13 2025
%H A144981 G. C. Greubel, <a href="/A144981/b144981.txt">Table of n, a(n) for n = 0..10000</a>
%H A144981 Christian Bingane and Charles Audet, <a href="https://arxiv.org/abs/2110.00036">The equilateral small octagon of maximal width</a>, arXiv:2110.00036 [math.MG], 2021.
%H A144981 Wikipedia, <a href="https://en.wikipedia.org/wiki/Exact_trigonometric_constants">Exact trigonometric constants</a>.
%H A144981 Wikipedia, <a href="https://en.wikipedia.org/wiki/Octagon">Octagon</a>.
%H A144981 <a href="/index/Al#algebraic_04">Index entries for algebraic numbers, degree 4</a>
%F A144981 Equals sqrt(2 + sqrt(2))/2 = sqrt(3.41421...)/2 = 1.8477759.../2.
%F A144981 Equals Hypergeometric2F1([11/16, 5/16], [1/2], 3/4) / 2. - _R. J. Mathar_, Oct 27 2008
%F A144981 Equals 2F1(-1/4,1/4;1/2;1/2) . - _R. J. Mathar_, Aug 31 2025
%e A144981 Equals 0.923879532511286756128183189396788286822416625863642486115097...
%p A144981 evalf(sqrt(2+sqrt(2))/2) ;
%t A144981 RealDigits[ Sqrt[2 + Sqrt[2]]/2, 10, 111][[1]] (* Or *) RealDigits[ Cos[Pi/8], 10, 111][[1]] (* _Robert G. Wilson v_ *)
%o A144981 (PARI) cos(Pi/8) \\ _Michel Marcus_, Dec 15 2015
%o A144981 (SageMath) numerical_approx(sqrt(2+sqrt(2))/2, digits=120) # _G. C. Greubel_, Sep 04 2022
%o A144981 (Python)
%o A144981 from math import isqrt # integer arithmetic, avoiding 10^(4N) in inner isqrt
%o A144981 def A144981_first(N=99): return [9] if N<2 else list(map(int,str(
%o A144981     isqrt(isqrt(100**(N+2)>>3)*10**(N-2)+100**N//2)))) # _M. F. Hasler_, Aug 13 2025
%Y A144981 Cf. A019863: cos(Pi/5), A010527: cos(Pi/6), A073052: cos(Pi/7), A019879: cos(Pi/9).
%K A144981 cons,easy,nonn,changed
%O A144981 0,1
%A A144981 _R. J. Mathar_, Sep 28 2008