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

%I A137218 #37 Feb 16 2025 08:33:07
%S A137218 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,
%T A137218 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,
%U A137218 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
%N A137218 Decimal expansion of the argument of -1 + 2*i.
%C A137218 Gives closed forms for many arctangent values:
%C A137218 arctan(2) = Pi - a, arctan(1/2) = a - Pi/2,
%C A137218 arctan(3) = a - Pi/4, arctan(1/3) = 3*Pi/4 - a,
%C A137218 arctan(7) = 7*Pi/4 - 2*a, arctan(1/7) = 2*a - 5*Pi/4,
%C A137218 arctan(4/3) = 2*a - Pi and arctan(3/4) = 3*Pi/2 - 2*a.
%C A137218 Dihedral angle in the dodecahedron (radians). - _R. J. Mathar_, Mar 24 2012
%C A137218 Larger interior angle (in radians) of a golden rhombus; A105199 is the smaller interior angle. - _Eric W. Weisstein_, Dec 17 2018
%H A137218 Rick L. Shepherd, <a href="/A137218/b137218.txt">Table of n, a(n) for n = 1..20000</a>
%H A137218 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GoldenRhombus.html">Golden Rhombus</a>
%H A137218 Wikipedia, <a href="http://en.wikipedia.org/wiki/Dodecahedron">Dodecahedron</a>
%H A137218 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%F A137218 Equals Pi - arctan(2) = A000796 - A105199 = 2*A195723.
%e A137218 2.0344439357957027354455779231...
%t A137218 RealDigits[Pi - ArcTan[2], 10, 120][[1]] (* _Harvey P. Dale_, Aug 08 2014 *)
%o A137218 (PARI)
%o A137218 default(realprecision, 120);
%o A137218 acos(-1/sqrt(5)) \\ or
%o A137218 arg(-1+2*I) \\ _Rick L. Shepherd_, Jan 26 2014
%Y A137218 Platonic solids' dihedral angles: A137914 (tetrahedron), A156546 (octahedron), A019669 (cube), A236367 (icosahedron). - _Stanislav Sykora_, Jan 23 2014
%Y A137218 Cf. A242723 (same in degrees).
%Y A137218 Cf. A105199 (smaller interior angle of the golden rhombus).
%K A137218 cons,nonn
%O A137218 1,1
%A A137218 Matt Rieckman (mjr162006(AT)yahoo.com), Mar 06 2008
%E A137218 Corrected a typo in the sequence Matt Rieckman (mjr162006(AT)yahoo.com), Feb 05 2010
%E A137218 More terms from _Rick L. Shepherd_, Jan 26 2014