A278924 - OEIS

A278924

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

4, 296, 36772, 1288688, 96641548, 26576092808, 8637277012172, 1079658805128928, 91770997994914276, 43591225139846360008

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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.

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