This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A105531 #58 Mar 22 2025 06:37:03
%S A105531 3,2,1,7,5,0,5,5,4,3,9,6,6,4,2,1,9,3,4,0,1,4,0,4,6,1,4,3,5,8,6,6,1,3,
%T A105531 1,9,0,2,0,7,5,5,2,9,5,5,5,7,6,5,6,1,9,1,4,3,2,8,0,3,0,5,9,3,5,6,7,5,
%U A105531 6,2,3,7,4,0,5,8,1,0,5,4,4,3,5,6,4,0,8,4,2,2,3,5,0,6,4,1,3,7,4,4,3,9,0,0,7
%N A105531 Decimal expansion of arctan(1/3).
%C A105531 arctan(1/3) + A073000 = 2*arctan(1/3) + A105533 = Pi/4.
%D A105531 John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 242.
%H A105531 Peter Bala, <a href="/A002117/a002117.pdf">New series for old functions</a>
%H A105531 Kunle Adegoke, <a href="http://arxiv.org/abs/1603.08097">Infinite arctangent sums involving Fibonacci and Lucas numbers</a>, arXiv:1603.08097 [math.NT], 2016.
%H A105531 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Machin-LikeFormulas.html">Machin-Like Formulas</a>
%H A105531 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F A105531 From _Peter Bala_, Feb 04 2015: (Start)
%F A105531 Equals (1/3)*Sum_{k >= 0} (-1)^k/((2*k + 1)*9^k).
%F A105531 Define a pair of integer sequences A(n) = 9^n*(2*n + 1)!/n! and B(n) = A(n)*Sum_{k = 0..n} (-1)^k/((2*k + 1)*9^k). Both sequences satisfy the same recurrence equation u(n) = (32*n + 20)*u(n-1) + 36*(2*n - 1)^2*u(n-2). From this observation we find the continued fraction expansion arctan(1/3) = (1/3)*(1 - 2/(54 + 36*3^2/(84 + 36*5^2/(116 + ... + 36*(2*n - 1)^2/((32*n + 20) + ...))))).
%F A105531 Equals (3/10) * Sum_{k >= 0} (2/5)^k/( (2*k + 1)*binomial(2*k,k) ).
%F A105531 Define a pair of integer sequences C(n) = 10^n*(2*n + 1)!/n! and D(n) = C(n)*Sum_{k = 0..n} (2/5)^k/( (2*k + 1)*binomial(2*k,k) ). Both sequences satisfy the same recurrence equation u(n) = (44*n + 20)*u(n-1) - 80*n*(2*n - 1)*u(n-2). From this observation we obtain the continued fraction expansion arctan(1/3) = (3/10)*( 1 + 4/(60 - 480/(108 - 1200/(152 - ... - 80*n*(2*n - 1)/((44*n + 20) - ...))))). (End)
%F A105531 Equals Sum_{k>=1} arctan(L(4*k+2)/F(4*k+2)^2) where L=A000032 and F=A000045. See also A033890 and A246453. - _Michel Marcus_, Mar 29 2016 [corrected by _Jason Yuen_, Jan 18 2025]
%F A105531 From _Amiram Eldar_, Aug 09 2020: (Start)
%F A105531 Equals Sum_{k>=2} arctan(1/(2*k^2)) = Sum_{k>=2} (-1)^k arctan(2/k^2).
%F A105531 Equals Integral_{x=1..2} 1/(x^2 + 1) dx. (End)
%F A105531 Equals Sum_{n>=0} arctan(1/F(2*n+5)) = Sum_{n>=0} (-1)^n arctan(F(2*n+1)) where F=A000045. - _Gleb Koloskov_, Oct 01 2021
%e A105531 0.3217505543966421934014046143...
%t A105531 RealDigits[ArcTan[1/3],10,120][[1]] (* _Harvey P. Dale_, Oct 28 2011 *)
%o A105531 (PARI) atan(1/3) \\ _Michel Marcus_, Mar 29 2016
%Y A105531 Cf. A003881 (Pi/4), A073000 (arctan(1/2)), A105533 (arctan(1/7)).
%K A105531 cons,nonn
%O A105531 0,1
%A A105531 Bryan Jacobs (bryanjj(AT)gmail.com), Apr 12 2005