

A293416


Decimal expansion of the minimum ripple factor for a ninthorder, 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
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OFFSET

0,1


COMMENTS

This is the smallest ripple factor (a constant) for which the prototype elements of the ninthorder 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 higherorder constants share the same initial terms, to greater length as the order increases.
There are simple radical expressions for the third and fifthorder constants (see formulas). Further, the thirdorder 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 higherorder constants or the limiting (infiniteorder) constant.


REFERENCES

M. Morgan, Reflectionless Filters, Norwood, MA: Artech House, pp. 129132, 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, ninthorder and the limiting "infiniteorder" 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



