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A278364 A sequence showing denominators in ratios tending to the constant Pi/4 = 0.785398163397448... .

%I A278364 #22 Dec 07 2016 11:23:40
%S A278364 5,375,46875,1640625,123046875,33837890625,10997314453125,
%T A278364 1374664306640625,116846466064453125,55502071380615234375
%N A278364 A sequence showing denominators in ratios tending to the constant Pi/4 = 0.785398163397448... .
%C A278364 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 A278364 Sanjar Abrarov, <a href="/A278364/b278364.txt">Table of n, a(n) for n = 1..49</a>
%H A278364 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 A278364 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 A278364 ------------------------------------------------
%e A278364 n    c(n)                   d(n)
%e A278364 ------------------------------------------------
%e A278364 1    4                      5
%e A278364 2    296                    375
%e A278364 3    36772                  46875
%e A278364 4    1288688                1640625
%e A278364 5    96641548               123046875
%e A278364 6    26576092808            33837890625
%e A278364 7    8637277012172          10997314453125
%e A278364 8    1079658805128928       1374664306640625
%e A278364 9    91770997994914276      116846466064453125
%e A278364 10   43591225139846360008   55502071380615234375
%e A278364 ------------------------------------------------
%e A278364 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 A278364 x := 1; (* argument x *)
%t A278364 M := 1; (* initial value for the integer M *)
%t A278364 n := 1; (* index *)
%t A278364 (* Note that arctan(1) = Pi/4 *)
%t A278364 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 A278364 sqn := {}; (* initiate the sequence *)
%t A278364 AppendTo[sqn, {"Index n", "Numerators", "Denominators"}];
%t A278364 While[M <= 20, AppendTo[sqn, {n, Numerator[atan], Denominator[atan]}];
%t A278364 {M = M + 2, n++}];
%t A278364 Print[MatrixForm[sqn]]
%Y A278364 Cf. A278924, A003881, A096954, A096955.
%K A278364 nonn,frac
%O A278364 1,1
%A A278364 _Sanjar Abrarov_, Dec 04 2016