%I #22 Dec 07 2016 11:23:40

%S 5,375,46875,1640625,123046875,33837890625,10997314453125,

%T 1374664306640625,116846466064453125,55502071380615234375

%N A sequence showing denominators 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. All denominators d(n) are divisible by 5.

%H Sanjar Abrarov, <a href="/A278364/b278364.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 Viéte'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. A278924, A003881, A096954, A096955.

%K nonn,frac

%O 1,1

%A _Sanjar Abrarov_, Dec 04 2016