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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 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 ------------------------------------------------ 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 June 18 16:37 EDT 2024. Contains 373482 sequences. (Running on oeis4.)