login
A105199
Decimal expansion of arctan(2).
14
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
Also <5_1> in Conway et al. (1999). - Eric W. Weisstein, Nov 06 2024
LINKS
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.
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
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. A137218 (larger interior angle of the golden rhombus).
Sequence in context: A201750 A198825 A073008 * A020791 A367910 A368184
KEYWORD
cons,nonn,changed
AUTHOR
Bryan Jacobs (bryanjj(AT)gmail.com), Apr 12 2005
EXTENSIONS
Offset corrected by R. J. Mathar, Apr 12 2010
STATUS
approved