A195790 -id:A195790 - cp's OEIS frontend

A195789 Decimal expansion of arctan(10).

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

Offset: 1

Clark Kimberling, Sep 24 2011

			arctan(10) = 1.471127674303734591852875571761730851855306377...

Index entries for transcendental numbers

Cf. A195790, A195791, A195792, A195793.

r = 10;
N[ArcTan[r], 100]
RealDigits[%]  (* A195789 *)
N[ArcCot[r], 100]
RealDigits[%]  (* A195790 *)
N[ArcSec[r], 100]
RealDigits[%]  (* A195791 *)
N[ArcCsc[r], 100]
RealDigits[%]  (* A195792 *)
RealDigits[ArcTan[10],10,120][[1]] (* Harvey P. Dale, Dec 27 2022 *)

atan(10) \\ Charles R Greathouse IV, Mar 24 2021

Equals arcsin(10/sqrt(101)) = arccos(1/sqrt(101)). - Amiram Eldar, Jul 11 2023

A105534 Decimal expansion of arctan 1/239.

Original entry on oeis.org

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

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

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)

			0.0041840760020747238645382149...

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

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

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 *)

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

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

cp's OEIS Frontend

A195789 Decimal expansion of arctan(10).

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