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A105199 Decimal expansion of arctan(2). 11
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 (list; constant; graph; refs; listen; history; text; internal format)
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

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..5000

G. Boros, 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]

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

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. A195693.

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

Sequence in context: A201750 A198825 A073008 * A020791 A086210 A085467

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

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Last modified March 20 21:49 EDT 2019. Contains 321352 sequences. (Running on oeis4.)