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. 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.
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
CROSSREFS
KEYWORD
AUTHOR
Bryan Jacobs (bryanjj(AT)gmail.com), Apr 12 2005
EXTENSIONS
Offset corrected by R. J. Mathar, Apr 12 2010
STATUS
approved