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%I A105534 #51 Feb 16 2025 08:32:57
%S A105534 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,
%T A105534 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,
%U A105534 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
%N A105534 Decimal expansion of arctan 1/239.
%C A105534 Comment from _Frank Ellermann_, Mar 01 2020: (Start)
%C A105534 8*A195790 - arctan( 1/239 ) - 4*arctan( 1/515 ) = Pi/4 (Meissel, Klingenstierna).
%C A105534 12*arctan( 1/18 ) + 8*arctan( 1/57 ) - 5*arctan( 1/239 ) = Pi/4 (Gauss). (End)
%H A105534 D. H. Lehmer, <a href="https://www.maa.org/sites/default/files/pdf/pubs/amm_supplements/Monthly_Reference_7.pdf">On Arccotangent Relations for π</a>, The American Mathematical Monthly, Vol. 45, No. 10 (Dec., 1938), pp. 657-664.
%H A105534 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Machin-LikeFormulas.html">Machin-Like Formulas</a>
%H A105534 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F A105534 4*A105532 - arctan(1/239) = Pi/4 (Machin's formula).
%F A105534 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
%e A105534 0.0041840760020747238645382149...
%t A105534 len = 103; n = RealDigits[N[ArcTan[1/239], len]]; PadLeft[First@ n, len + Abs@ Last@ n] (* _Michael De Vlieger_, Sep 14 2015 *)
%t A105534 Join[{0,0},RealDigits[ArcTan[1/239],10,120][[1]]] (* _Harvey P. Dale_, Apr 29 2016 *)
%o A105534 (PARI) atan(1/239) \\ _Michel Marcus_, Sep 24 2014
%Y A105534 Cf. A003881 (Pi/4), A021243 (1/239), A105532 (arctan 1/5), A195790 (arccot 10).
%K A105534 cons,nonn
%O A105534 0,3
%A A105534 Bryan Jacobs (bryanjj(AT)gmail.com), Apr 12 2005

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