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A293416 Decimal expansion of the minimum ripple factor for a ninth-order, reflectionless, Chebyshev filter. 8
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, 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, 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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

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.

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

G. C. Greubel, Table of n, a(n) for n = 0..10000

FORMULA

Equals sqrt(exp(4*arctanh(exp(-2*9*arcsinh(sqrt(1/2*sin(Pi/9)tan(Pi/9))))))-1).

EXAMPLE

0.2192047733...

MATHEMATICA

RealDigits[Sqrt[Exp[4 ArcTanh[Exp[-2*9*ArcSinh[Sqrt[1/2*Sin[Pi/9] Tan[Pi/9]]]]]] - 1], 10, 100][[1]]

PROG

(PARI) sqrt(exp(4*atanh(exp(-2*9*asinh(sqrt(1/2*sin(Pi/9)*tan(Pi/9))))))-1) \\ Michel Marcus, Oct 16 2017

(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

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: A133399 A128751 A129168 * A194555 A024578 A030327

Adjacent sequences:  A293413 A293414 A293415 * A293417 A293418 A293419

KEYWORD

cons,easy,nonn

AUTHOR

Matthew A. Morgan, Oct 15 2017

STATUS

approved

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Last modified May 28 12:18 EDT 2020. Contains 334681 sequences. (Running on oeis4.)