A105532 - OEIS

%I #14 Nov 20 2024 23:40:57

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

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

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

%N Decimal expansion of arctan(1/5).

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

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>

%F 4*arctan(1/5) - A105534 = Pi/4 (Machin's formula).

%F From _Amiram Eldar_, Aug 04 2020: (Start)

%F Equals Sum_{k>=0} (-1)^k/((2*k+1) * 5^(2*k+1)) = Sum_{k>=0} (-1)^k/A072172(k).

%F Equals Sum_{k>=1} arctan(1/(2*(k+2)^2)). (End)

%e 0.197395559849880758370049765194790293447585103787852101517688940241033969...

%t RealDigits[ArcTan[1/5], 10, 100][[1]] (* _Amiram Eldar_, Aug 04 2020 *)

%o (PARI) atan(1/5) \\ _Michel Marcus_, Sep 24 2014

%Y Cf. A003881 (Pi/4), A072172, A105534 (arctan 1/239).

%K cons,nonn

%O 0,2

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

%E Corrected position of decimal point in example. - _R. J. Mathar_, Feb 05 2009