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A105534 Decimal expansion of arctan 1/239. 1

%I #48 Mar 22 2024 07:06:41

%S 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 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 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 Decimal expansion of arctan 1/239.

%C Comment from _Frank Ellermann_, Mar 01 2020: (Start)

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

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

%H 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Machin-LikeFormulas.html">Machin-Like Formulas</a>

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

%F 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 0.0041840760020747238645382149...

%t len = 103; n = RealDigits[N[ArcTan[1/239], len]]; PadLeft[First@ n, len + Abs@ Last@ n] (* _Michael De Vlieger_, Sep 14 2015 *)

%t Join[{0,0},RealDigits[ArcTan[1/239],10,120][[1]]] (* _Harvey P. Dale_, Apr 29 2016 *)

%o (PARI) atan(1/239) \\ _Michel Marcus_, Sep 24 2014

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

%K cons,nonn,changed

%O 0,3

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

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Last modified March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)