A105534 Decimal expansion of arctan 1/239.

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

Offset: 0

Author

Bryan Jacobs (bryanjj(AT)gmail.com), Apr 12 2005

Keywords

• cons
• nonn

Comments

Comment from Frank Ellermann, Mar 01 2020: (Start)

8*A195790 - arctan( 1/239 ) - 4*arctan( 1/515 ) = Pi/4 (Meissel, Klingenstierna).

12*arctan( 1/18 ) + 8*arctan( 1/57 ) - 5*arctan( 1/239 ) = Pi/4 (Gauss). (End)

Examples

			0.0041840760020747238645382149...
		

Links

• D. H. Lehmer, On Arccotangent Relations for π, The American Mathematical Monthly, Vol. 45, No. 10 (Dec., 1938), pp. 657-664.
• Eric Weisstein's World of Mathematics, Machin-Like Formulas
• Index entries for transcendental numbers

Crossrefs

Cf. A003881 (Pi/4), A021243 (1/239), A105532 (arctan 1/5), A195790 (arccot 10).

Programs

Mathematica

len = 103; n = RealDigits[N[ArcTan[1/239], len]]; PadLeft[First@ n, len + Abs@ Last@ n] (* Michael De Vlieger, Sep 14 2015 *)
Join[{0,0},RealDigits[ArcTan[1/239],10,120][[1]]] (* Harvey P. Dale, Apr 29 2016 *)

PARI

atan(1/239) \\ Michel Marcus, Sep 24 2014

Formula

4*A105532 - arctan(1/239) = Pi/4 (Machin's formula).

arctan(1/239) = Sum_{n >= 1} i/(n*P(n, 239*i)*P(n-1, 239*i)) = 1/239 - 1/40955996 + 1/8773020079176 - 1/1948832181801673304 + 4/1753293766205137615850855 - ..., where i = sqrt(-1) and P(n, x) denotes the n-th Legendre polynomial. - Peter Bala, Mar 21 2024