A293417 Decimal expansion of the minimum ripple factor for a reflectionless, Chebyshev filter, in the limit where the order approaches infinity.

2, 1, 9, 4, 8, 6, 9, 3, 0, 8, 7, 6, 8, 1, 3, 9, 1, 6, 8, 9, 4, 5, 8, 8, 3, 4, 4, 8, 7, 6, 6, 0, 7, 1, 7, 9, 4, 3, 0, 9, 2, 1, 3, 3, 3, 1, 6, 8, 8, 3, 8, 7, 4, 1, 9, 4, 1, 9, 8, 0, 8, 8, 6, 1, 2, 7, 5, 1, 0, 0, 4, 6, 9, 4, 6, 8, 7, 0, 8, 2, 4, 5, 2, 8, 3, 7, 3, 5, 5, 2, 5, 1, 5, 5, 2, 4, 0, 5, 0, 7, 4, 4, 7, 5, 9, 6, 8, 7

Offset

0,1

Comments

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.

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.

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.

References

M. Morgan, Reflectionless Filters, Norwood, MA: Artech House, pp. 129-132, January 2017.

Links

Iain Fox, Table of n, a(n) for n = 0..20000

Formula

Equals sqrt(exp(4*arctanh(exp(-Pi*sqrt(2))))-1).

Example

0.2194869308...

Mathematica

RealDigits[Sqrt[Exp[4 ArcTanh[Exp[-(Pi Sqrt[2])]]] - 1], 10, 100][[1]]

Prog

(PARI) sqrt(exp(4*atanh(exp(-Pi*sqrt(2))))-1) \\ Michel Marcus, Oct 15 2017
(Magma) R:= RealField(); Sqrt(Exp(4*Argtanh(Exp(-Pi(R)*Sqrt(2))))-1); // G. C. Greubel, Feb 16 2018

Crossrefs

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.

Sequence in context: A197411 A124905 A371939 * A295737 A021086 A372804

Adjacent sequences: A293414 A293415 A293416 * A293418 A293419 A293420

Keyword

cons,easy,nonn

Author

Matthew A. Morgan, Oct 15 2017

Status

approved