A137218 Decimal expansion of the argument of -1 + 2*i.

2, 0, 3, 4, 4, 4, 3, 9, 3, 5, 7, 9, 5, 7, 0, 2, 7, 3, 5, 4, 4, 5, 5, 7, 7, 9, 2, 3, 1, 0, 0, 9, 6, 5, 8, 4, 4, 1, 2, 7, 1, 2, 1, 7, 5, 3, 9, 7, 3, 6, 7, 3, 1, 7, 4, 2, 9, 8, 4, 0, 5, 3, 8, 4, 8, 7, 4, 1, 0, 6, 0, 6, 7, 3, 0, 8, 8, 4, 6, 2, 0, 4, 6, 1, 4, 6, 1, 7, 6, 9, 6, 6, 5, 5, 9, 4, 6, 4, 2, 6, 5, 4, 7, 6, 0

Offset: 1

Author

Matt Rieckman (mjr162006(AT)yahoo.com), Mar 06 2008

Keywords

• cons
• nonn

Comments

Gives closed forms for many arctangent values:

arctan(2) = Pi - a, arctan(1/2) = a - Pi/2,

arctan(3) = a - Pi/4, arctan(1/3) = 3*Pi/4 - a,

arctan(7) = 7*Pi/4 - 2*a, arctan(1/7) = 2*a - 5*Pi/4,

arctan(4/3) = 2*a - Pi and arctan(3/4) = 3*Pi/2 - 2*a.

Dihedral angle in the dodecahedron (radians). - R. J. Mathar, Mar 24 2012

Larger interior angle (in radians) of a golden rhombus; A105199 is the smaller interior angle. - Eric W. Weisstein, Dec 17 2018

Examples

			2.0344439357957027354455779231...
		

Links

• Rick L. Shepherd, Table of n, a(n) for n = 1..20000
• Eric Weisstein's World of Mathematics, Golden Rhombus
• Wikipedia, Dodecahedron
• Index entries for transcendental numbers.

Crossrefs

Platonic solids' dihedral angles: A137914 (tetrahedron), A156546 (octahedron), A019669 (cube), A236367 (icosahedron). - Stanislav Sykora, Jan 23 2014

Cf. A242723 (same in degrees).

Cf. A105199 (smaller interior angle of the golden rhombus).

Programs

Mathematica

RealDigits[Pi - ArcTan[2], 10, 120][[1]] (* Harvey P. Dale, Aug 08 2014 *)

PARI

default(realprecision, 120);
acos(-1/sqrt(5)) \\ or
arg(-1+2*I) \\ Rick L. Shepherd, Jan 26 2014

Formula

Equals Pi - arctan(2) = A000796 - A105199 = 2*A195723.

Extensions

Corrected a typo in the sequence Matt Rieckman (mjr162006(AT)yahoo.com), Feb 05 2010

More terms from Rick L. Shepherd, Jan 26 2014