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

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, 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, 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

Offset: 0

Author

R. J. Mathar, Sep 28 2008

Keywords

• cons
• easy
• nonn

Comments

Also the real part of i^(1/4). - Stanislav Sykora, Apr 25 2012

Width of a regular octagon of unit diameter. See Bingane and Audet. - Michel Marcus, Oct 04 2021

Minimal polynomial 8x^4 - 8x^2 + 1. - Charles R Greathouse IV, Oct 30 2023

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

Examples

			Equals 0.923879532511286756128183189396788286822416625863642486115097...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..10000
• Christian Bingane and Charles Audet, The equilateral small octagon of maximal width, arXiv:2110.00036 [math.MG], 2021.
• Wikipedia, Exact trigonometric constants.
• Wikipedia, Octagon.
• Index entries for algebraic numbers, degree 4

Crossrefs

Cf. A019863: cos(Pi/5), A010527: cos(Pi/6), A073052: cos(Pi/7), A019879: cos(Pi/9).

Programs

Maple

evalf(sqrt(2+sqrt(2))/2) ;

Mathematica

RealDigits[ Sqrt[2 + Sqrt[2]]/2, 10, 111][[1]] (* Or *) RealDigits[ Cos[Pi/8], 10, 111][[1]] (* Robert G. Wilson v *)

PARI

cos(Pi/8) \\ Michel Marcus, Dec 15 2015

Python

from math import isqrt # integer arithmetic, avoiding 10^(4N) in inner isqrt
def A144981_first(N=99): return [9] if N<2 else list(map(int,str(
    isqrt(isqrt(100**(N+2)>>3)*10**(N-2)+100**N//2)))) # M. F. Hasler, Aug 13 2025

SageMath

numerical_approx(sqrt(2+sqrt(2))/2, digits=120) # G. C. Greubel, Sep 04 2022

Formula

Equals sqrt(2 + sqrt(2))/2 = sqrt(3.41421...)/2 = 1.8477759.../2.

Equals Hypergeometric2F1([11/16, 5/16], [1/2], 3/4) / 2. - R. J. Mathar, Oct 27 2008