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A105199
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Decimal expansion of arctan(2).
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13
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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
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OFFSET
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1,4
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COMMENTS
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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
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LINKS
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FORMULA
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Equals Sum_{k>=1} arctan(8k/(4k^4+5)). [Boros and Moll, from R. J. Mathar, Apr 12 2010]
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
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EXAMPLE
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1.107148717794090503017065460...
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MATHEMATICA
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PROG
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(PARI)
default(realprecision, 120);
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CROSSREFS
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Cf. A137218 (larger interior angle of the golden rhombus).
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KEYWORD
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AUTHOR
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Bryan Jacobs (bryanjj(AT)gmail.com), Apr 12 2005
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EXTENSIONS
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STATUS
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approved
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