|
%I #40 Oct 06 2023 16:05:02
%S 1,1,0,7,1,4,8,7,1,7,7,9,4,0,9,0,5,0,3,0,1,7,0,6,5,4,6,0,1,7,8,5,3,7,
%T 0,4,0,0,7,0,0,4,7,6,4,5,4,0,1,4,3,2,6,4,6,6,7,6,5,3,9,2,0,7,4,3,3,7,
%U 1,0,3,3,8,9,7,7,3,6,2,7,9,4,0,1,3,4,1,7,1,2,8,6,8,6,1,7,0,6,4,1,4,3,4,5,4
%N Decimal expansion of arctan(2).
%C arctan(2) + A073000 = Pi/2.
%C arctan(2) is the (minimal) central angle of a regular icosahedron, which is the platonic solid having 20 faces and 12 vertices. The (minimal) central angle is AOB, where A and B are any neighboring pair of vertices and O is the center. To evaluate AOB, it is helpful to start with 12 vertices: (0,c*t,d), (d,0,c*t), (c*t,d,0) where c=(1 or -1) and d=(1 or -1) and t is the golden ratio, (1+sqrt(5))/2. For neighboring vertices, one can select (t,1,0) and (0,t,1). - _Clark Kimberling_, Feb 10 2009
%C Lesser interior angle (in radians) of a golden rhombus; i.e., either of the angles bisected by the longer diagonal. A137218 is the greater interior angle. - _Rick L. Shepherd_, Apr 10 2017
%C The apex angle in the isosceles triangle that is the triangle with angles A, B and C in which the maximum values of sin(A) + sin(B)*sin(C) is attained. The maximum value is phi (A001622) (Rabinowitz, 2007). - _Amiram Eldar_, Aug 04 2022
%H G. C. Greubel, <a href="/A105199/b105199.txt">Table of n, a(n) for n = 1..5000</a>
%H G. Boros and V. Moll, <a href="http://www.mat.utfsm.cl/scientia/archivos/vol11/Art2.pdf">Sums of arctangents and some formulas of Ramanujan</a>, Scientia Ser. A Math. Sci 11 (2005) 13-24 [<a href="http://www.ams.org/mathscinet-getitem?mr=2196063">MR2196063</a>] eq. (2.11). [From _R. J. Mathar_, Apr 12 2010]
%H Stanley Rabinowitz, <a href="https://cms.math.ca/publications/crux/issue?volume=33&issue=8">Problem 3297</a>, Crux Mathematicorum, Vol. 33, No. 8 (2007), pp. 486 and 488; <a href="https://cms.math.ca/publications/crux/issue?volume=34&issue=8">Solution to Problem 3297</a> by D. J. Smeenk, ibid., Vol. 34, No. 8 (2008), pp. 499-500.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GoldenRhombus.html">Golden Rhombus</a>.
%F Equals Sum_{k>=1} arctan(8k/(4k^4+5)). [Boros and Moll, from _R. J. Mathar_, Apr 12 2010]
%F Equals 2*A195693. - _Rick L. Shepherd_, Apr 10 2017
%F Equals arcsin(2/sqrt(5)) = arccos(1/sqrt(5)). - _Amiram Eldar_, Aug 04 2022
%F Equals 2 - log(5) + (Integral_{x=0..2} log(1 + x^2) dx)/2. - _Vaclav Kotesovec_, Oct 06 2023
%e 1.107148717794090503017065460...
%t RealDigits[ArcTan[2], 10, 105][[1]] (* _Indranil Ghosh_, Apr 10 2017 *)
%o (PARI)
%o default(realprecision, 120);
%o atan(2) \\ _Rick L. Shepherd_, Apr 10 2017
%Y Cf. A001622, A195693.
%Y Cf. A137218 (larger interior angle of the golden rhombus).
%K cons,nonn
%O 1,4
%A Bryan Jacobs (bryanjj(AT)gmail.com), Apr 12 2005
%E Offset corrected by _R. J. Mathar_, Apr 12 2010
|
