OFFSET
1,1
COMMENTS
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.
LINKS
Sanjar Abrarov, Table of n, a(n) for n = 1..49
S. M. Abrarov and B. M. Quine, A generalized Viéte's-like formula for pi with rapid convergence, arXiv:1610.07713 [math.GM], (2016).
FORMULA
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))
EXAMPLE
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n c(n) d(n)
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1 4 5
2 296 375
3 36772 46875
4 1288688 1640625
5 96641548 123046875
6 26576092808 33837890625
7 8637277012172 10997314453125
8 1079658805128928 1374664306640625
9 91770997994914276 116846466064453125
10 43591225139846360008 55502071380615234375
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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.
MATHEMATICA
x := 1; (* argument x *)
M := 1; (* initial value for the integer M *)
n := 1; (* index *)
(* Note that arctan(1) = Pi/4 *)
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}];
sqn := {}; (* initiate the sequence *)
AppendTo[sqn, {"Index n", "Numerators", "Denominators"}];
While[M <= 20, AppendTo[sqn, {n, Numerator[atan], Denominator[atan]}];
{M = M + 2, n++}];
Print[MatrixForm[sqn]]
CROSSREFS
KEYWORD
nonn,frac
AUTHOR
Sanjar Abrarov, Dec 04 2016
STATUS
approved