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A168229 Decimal expansion of arctan(sqrt(7)). 5

%I #52 Sep 08 2022 08:45:48

%S 1,2,0,9,4,2,9,2,0,2,8,8,8,1,8,8,8,1,3,6,4,2,1,3,3,0,1,5,3,1,9,0,8,4,

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

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

%N Decimal expansion of arctan(sqrt(7)).

%C This constant is the least x > 0 satisfying cos(4*x) = (cos x)^2. - _Clark Kimberling_, Oct 15 2011

%C An identity resembling Machin's Pi/4 = arctan(1/1) = 4*arctan(1/5) - arctan(1/239) is arctan(sqrt(7)/1) = 5*arctan(sqrt(7)/11) + 2*arctan(sqrt(7)/181), which can also be expressed as arcsin(sqrt(7/2^3)) = 5*arcsin(sqrt(7/2^7)) + 2*arcsin(sqrt(7/2^15)) (cf. A038198). - _Joerg Arndt_, Nov 09 2012

%H G. C. Greubel, <a href="/A168229/b168229.txt">Table of n, a(n) for n = 1..5000</a>

%H Kunle Adegoke, <a href="http://arxiv.org/abs/1603.08097">Infinite arctangent sums involving Fibonacci and Lucas numbers</a>, arXiv:1603.08097 [math.NT], 2016.

%H Djurdje Cvijovic, <a href="http://arxiv.org/abs/0911.3773">A dilogarithmic integral arising in quantum field theory</a>, arXiv:0911.3773 [math.CA], 2009.

%F Smallest positive solution of cos(x) + sqrt(1 + cos^2(x)) = sqrt(2). - _Geoffrey Caveney_, Apr 24 2014

%F Equals Sum_{k >= 1} atan(5*sqrt(7)*F(4k-1)/L(2*(4k-1))) where L=A000032 and F=A000045. See also A033891. - _Michel Marcus_, Mar 29 2016

%F Equals arccos(1/(2*sqrt(2))). - _Amiram Eldar_, May 28 2021

%e arctan(sqrt(7)) = 1.209429202888189... .

%t RealDigits[ArcTan[Sqrt[7]], 10, 50][[1]] (* _G. C. Greubel_, Nov 18 2017 *)

%o (PARI) atan(sqrt(7)) \\ _Michel Marcus_, Mar 11 2013

%o (Magma) [Arctan(Sqrt(7))]; // _G. C. Greubel_, Nov 18 2017

%Y Cf. A000032, A000045, A033891, A038198, A195699.

%K cons,nonn

%O 1,2

%A _Jonathan Vos Post_, Nov 20 2009

%E More digits from _R. J. Mathar_, Dec 06 2009

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