%I #26 Dec 07 2016 11:24:26
%S 4,296,36772,1288688,96641548,26576092808,8637277012172,
%T 1079658805128928,91770997994914276,43591225139846360008
%N A sequence showing numerators in ratios tending to the constant Pi/4 = 0.785398163397448... .
%C The ratios c(n)/d(n) rapidly tend to the constant Pi/4 = 0.785398163397448... with increasing index n: abs(Pi/4 - c(1)/d(1)) > abs(Pi/4 - c(2)/d(2)) > abs(Pi/4 - c(3)/d(3)) > abs(Pi/4 - c(4)/d(4)) ..., where c(n) = A278924(n) and d(n) = A278364(n) are even and odd positive integers, respectively.
%H Sanjar Abrarov, <a href="/A278924/b278924.txt">Table of n, a(n) for n = 1..49</a>
%H S. M. Abrarov and B. M. Quine, <a href="https://arxiv.org/abs/1610.07713">A generalized ViƩte's-like formula for pi with rapid convergence</a>, arXiv:1610.07713 [math.GM], (2016).
%F arctan(1) = I*lim_{M -> inf}Sum_{m = 1..floor(M/2) + 1}(1/(2*m - 1))*(1/(1 + 2*I)^(2*m - 1) - 1/(1 - 2*I)^(2*m - 1))
%e ------------------------------------------------
%e n c(n) d(n)
%e ------------------------------------------------
%e 1 4 5
%e 2 296 375
%e 3 36772 46875
%e 4 1288688 1640625
%e 5 96641548 123046875
%e 6 26576092808 33837890625
%e 7 8637277012172 10997314453125
%e 8 1079658805128928 1374664306640625
%e 9 91770997994914276 116846466064453125
%e 10 43591225139846360008 55502071380615234375
%e ------------------------------------------------
%e At n = 6 the ratio c(6)/d(6) = 26576092808/33837890625 is close to Pi/4. However, at n = 10 the ratio c(10)/d(10) = 43591225139846360008/55502071380615234375 becomes more closer to Pi/4.
%t x := 1;(* argument x *)
%t M := 1;(* initial value for the integer M *)
%t n := 1; (* index *)
%t (* Note that arctan(1) = Pi/4 *)
%t atan := I*Sum[(1/(2*m - 1))*(1/(1 + 2*(I/x))^(2*m - 1) - 1/(1 - 2*(I/x))^(2*m - 1)),{m, 1, Floor[M/2] + 1}];
%t sqn := {};(* initiate the sequence *)
%t AppendTo[sqn,{"Index n", "Numerators", "Denominators"}];
%t While[M <= 20, AppendTo[sqn,{n, Numerator[atan], Denominator[atan]}];
%t {M = M + 2, n++}];
%t Print[MatrixForm[sqn]]
%Y Cf. A278364, A003881, A096954, A096955.
%K nonn,frac
%O 1,1
%A _Sanjar Abrarov_, Dec 01 2016