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A278364 A sequence showing denominators in ratios tending to the constant Pi/4 = 0.785398163397448... . 2
5, 375, 46875, 1640625, 123046875, 33837890625, 10997314453125, 1374664306640625, 116846466064453125, 55502071380615234375 (list; graph; refs; listen; history; text; internal format)
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 ViƩte'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

------------------------------------------------

n    c(n)                   d(n)

------------------------------------------------

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

------------------------------------------------

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

Cf. A278924, A003881, A096954, A096955.

Sequence in context: A098038 A354831 A072172 * A214008 A208094 A273397

Adjacent sequences:  A278361 A278362 A278363 * A278365 A278366 A278367

KEYWORD

nonn,frac

AUTHOR

Sanjar Abrarov, Dec 04 2016

STATUS

approved

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Last modified October 2 23:51 EDT 2022. Contains 357230 sequences. (Running on oeis4.)