cp's OEIS Frontend

A293882 Continued fraction expansion of the minimum ripple factor for a reflectionless, Chebyshev filter, in the limit where the order approaches infinity.

%I A293882 #26 Aug 25 2024 16:39:11
%S A293882 0,4,1,1,3,1,22,1,3,3,1,1,1,13,10,3,4,2,7,1,4,6,2,4,1,1,6,2,1,2,1,1,2,
%T A293882 3,42,3,6,3,2,1,1,1,2,2,8,2,4,1,2,3,1,1,1,2,5,8,3,1,1,3,2,3,2,11,1,3,
%U A293882 6,6,1,1,3,1,1,103,2,2,2,3,2,44,2,1,1,2,1,5,1,9,1,1,5,1,1,7,1,2,2,1,1,1,5,1,1,1,4,45
%N A293882 Continued fraction expansion of the minimum ripple factor for a reflectionless, Chebyshev filter, in the limit where the order approaches infinity.
%C A293882 This is the smallest ripple factor (a constant) for which the prototype elements of the generalized reflectionless filter topology (see Morgan, 2017) needs no negative elements, where the order of the filter approaches infinity. 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 A293882 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 A293882 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 A293882 M. Morgan, Reflectionless Filters, Norwood, MA: Artech House, pp. 129-132, January 2017.
%H A293882 G. C. Greubel, <a href="/A293882/b293882.txt">Table of n, a(n) for n = 0..9999</a>
%e A293882 1/(4 + 1/(1 + 1/(1 + 1/(3 + 1/(1 + 1/(22 + 1/(1 + 1/(3 + 1/(3 +...
%t A293882 ContinuedFraction[Sqrt[Exp[4 ArcTanh[Exp[-(Pi Sqrt[2])]]] - 1],130]
%o A293882 (Magma) R:= RealField();ContinuedFraction(Sqrt(Exp( 4*Argtanh(Exp (-(Pi(R)*Sqrt(2))))) - 1)); // _Michel Marcus_, Feb 17 2018
%o A293882 (PARI) contfrac(sqrt(exp(4*atanh(exp(-Pi*sqrt(2)))) - 1)) \\ _Michel Marcus_, Feb 17 2018
%Y A293882 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 A293882 cofr,easy,nonn
%O A293882 0,2
%A A293882 _Matthew A. Morgan_, Oct 18 2017
%E A293882 Offset changed by _Andrew Howroyd_, Aug 10 2024