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

1,4

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

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]

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, Golden Rhombus.

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

Example

1.107148717794090503017065460...

Mathematica

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

Prog

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

Crossrefs

Cf. A001622, A195693.

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

Sequence in context: A201750 A198825 A073008 * A020791 A367910 A368184

Adjacent sequences: A105196 A105197 A105198 * A105200 A105201 A105202

Keyword

cons,nonn

Author

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

Extensions

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

Status

approved