cp's OEIS Frontend

A293416 Decimal expansion of the minimum ripple factor for a ninth-order, reflectionless, Chebyshev filter.

%I A293416 #37 Sep 08 2022 08:46:19
%S A293416 2,1,9,2,0,4,7,7,3,3,7,2,5,0,6,0,7,5,8,3,0,3,5,7,9,9,3,1,3,5,3,8,6,6,
%T A293416 4,7,9,9,8,5,3,2,7,6,5,4,6,2,4,2,8,4,7,1,7,6,8,4,5,6,0,3,0,7,8,4,7,0,
%U A293416 5,9,2,6,2,1,8,7,3,7,9,3,5,0,7,3,2,9,2,3,9,0,5,9,8,8,1,4,8,0,4,5,2,7,0,6,4,2,3,7
%N A293416 Decimal expansion of the minimum ripple factor for a ninth-order, reflectionless, Chebyshev filter.
%C A293416 This is the smallest ripple factor (a constant) for which the prototype elements of the ninth-order generalized reflectionless filter topology (see Morgan, 2017) needs no negative elements. It is also the ripple factor for which the first two and last two Chebyshev prototype parameters (of the canonical ladder, or Cauer, topology) are equal.
%C A293416 Other related sequences in the OEIS are the decimal and continued fraction expansions of the limiting ripple factors for third, fifth, seventh, and ninth order, as well as for the limiting case where the order diverges to infinity. As these ripple factors do approach a common limit very quickly, the sequences for the fifth- and higher-order constants share the same initial terms, to greater length as the order increases.
%C A293416 There are simple radical expressions for the third- and fifth-order constants (see formulas). Further, the third-order constant is a quadratic irrational, thus having a repeating continued fraction expansion. I do not know if such simple expressions or patterns exist for the higher-order constants or the limiting (infinite-order) constant.
%D A293416 M. Morgan, Reflectionless Filters, Norwood, MA: Artech House, pp. 129-132, January 2017.
%H A293416 G. C. Greubel, <a href="/A293416/b293416.txt">Table of n, a(n) for n = 0..10000</a>
%F A293416 Equals sqrt(exp(4*arctanh(exp(-2*9*arcsinh(sqrt(1/2*sin(Pi/9)tan(Pi/9))))))-1).
%e A293416 0.2192047733...
%t A293416 RealDigits[Sqrt[Exp[4 ArcTanh[Exp[-2*9*ArcSinh[Sqrt[1/2*Sin[Pi/9] Tan[Pi/9]]]]]] - 1], 10,100][[1]]
%o A293416 (PARI) sqrt(exp(4*atanh(exp(-2*9*asinh(sqrt(1/2*sin(Pi/9)*tan(Pi/9))))))-1) \\ _Michel Marcus_, Oct 16 2017
%o A293416 (Magma) R:= RealField(); Sqrt(Exp(4*Argtanh(Exp(-2*9*Argsinh(Sqrt(1/2* Sin(Pi(R)/9)*Tan(Pi(R)/9))))))-1); // _G. C. Greubel_, Feb 16 2018
%Y A293416 Decimal expansions (A020784, A293409, A293415, A293416, A293417) and continued fractions (A040021, A293768, A293769, A293770, A293882) for third-, fifth-, seventh-, ninth-order and the limiting "infinite-order" constant, respectively.
%K A293416 cons,easy,nonn
%O A293416 0,1
%A A293416 _Matthew A. Morgan_, Oct 15 2017