cp's OEIS Frontend

A105199 Decimal expansion of arctan(2).

%I A105199 #49 Nov 20 2024 23:36:49
%S A105199 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 A105199 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 A105199 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 A105199 Decimal expansion of arctan(2).
%C A105199 arctan(2) + A073000 = Pi/2.
%C A105199 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 A105199 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 A105199 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
%C A105199 Also <5_1> in Conway et al. (1999). - _Eric W. Weisstein_, Nov 06 2024
%H A105199 G. C. Greubel, <a href="/A105199/b105199.txt">Table of n, a(n) for n = 1..5000</a>
%H A105199 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 A105199 J. H. Conway, C. Radin, L. and Sadun, On Angles Whose Squared Trigonometric Functions Are Rational. Discr. Computat. Geom., Vol. 22, (1999), pp. 321-332.
%H A105199 Stanley Rabinowitz, <a href="https://cms.math.ca/publications/crux/issue?volume=33&amp;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&amp;issue=8">Solution to Problem 3297</a> by D. J. Smeenk, ibid., Vol. 34, No. 8 (2008), pp. 499-500.
%H A105199 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DehnInvariant.html">Dehn Invariant</a>.
%H A105199 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GoldenRhombus.html">Golden Rhombus</a>.
%H A105199 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F A105199 Equals Sum_{k>=1} arctan(8k/(4k^4+5)). [Boros and Moll, from _R. J. Mathar_, Apr 12 2010]
%F A105199 Equals 2*A195693. - _Rick L. Shepherd_, Apr 10 2017
%F A105199 Equals arcsin(2/sqrt(5)) = arccos(1/sqrt(5)). - _Amiram Eldar_, Aug 04 2022
%F A105199 Equals 2 - log(5) + (Integral_{x=0..2} log(1 + x^2) dx)/2. - _Vaclav Kotesovec_, Oct 06 2023
%F A105199 Equals 3*A197292 = A197376/2. - _Hugo Pfoertner_, Nov 06 2024
%e A105199 1.107148717794090503017065460...
%t A105199 RealDigits[ArcTan[2], 10, 105][[1]] (* _Indranil Ghosh_, Apr 10 2017 *)
%o A105199 (PARI)
%o A105199 default(realprecision, 120);
%o A105199 atan(2) \\ _Rick L. Shepherd_, Apr 10 2017
%Y A105199 Cf. A001622, A195693, A197292, A197376.
%Y A105199 Cf. A137218 (larger interior angle of the golden rhombus).
%K A105199 cons,nonn
%O A105199 1,4
%A A105199 Bryan Jacobs (bryanjj(AT)gmail.com), Apr 12 2005
%E A105199 Offset corrected by _R. J. Mathar_, Apr 12 2010