A105199 Decimal expansion of arctan(2).

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, 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, 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

Offset: 1

Author

Bryan Jacobs (bryanjj(AT)gmail.com), Apr 12 2005

Keywords

• cons
• nonn

Comments

arctan(2) + A073000 = Pi/2.

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

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

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

Also <5_1> in Conway et al. (1999). - Eric W. Weisstein, Nov 06 2024

Examples

			1.107148717794090503017065460...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..5000
• G. Boros and V. Moll, Sums of arctangents and some formulas of Ramanujan, Scientia Ser. A Math. Sci 11 (2005) 13-24 [MR2196063] eq. (2.11). [From _R. J. Mathar_, Apr 12 2010]
• 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.
• Stanley Rabinowitz, Problem 3297, Crux Mathematicorum, Vol. 33, No. 8 (2007), pp. 486 and 488; Solution to Problem 3297 by D. J. Smeenk, ibid., Vol. 34, No. 8 (2008), pp. 499-500.
• Eric Weisstein's World of Mathematics, Dehn Invariant.
• Eric Weisstein's World of Mathematics, Golden Rhombus.
• Index entries for transcendental numbers

Crossrefs

Cf. A001622, A195693, A197292, A197376.

Cf. A137218 (larger interior angle of the golden rhombus).

Programs

Mathematica

RealDigits[ArcTan[2], 10, 105][[1]] (* Indranil Ghosh, Apr 10 2017 *)

PARI

default(realprecision, 120);
atan(2) \\ Rick L. Shepherd, Apr 10 2017

Formula

Equals Sum_{k>=1} arctan(8k/(4k^4+5)). [Boros and Moll, from R. J. Mathar, Apr 12 2010]

Equals 2*A195693. - Rick L. Shepherd, Apr 10 2017

Equals arcsin(2/sqrt(5)) = arccos(1/sqrt(5)). - Amiram Eldar, Aug 04 2022

Equals 2 - log(5) + (Integral_{x=0..2} log(1 + x^2) dx)/2. - Vaclav Kotesovec, Oct 06 2023

Equals 3*A197292 = A197376/2. - Hugo Pfoertner, Nov 06 2024

Extensions

Offset corrected by R. J. Mathar, Apr 12 2010