A293409 -id:A293409 - cp's OEIS frontend

A293415 Decimal expansion of the minimum ripple factor for a seventh-order, reflectionless, Chebyshev filter.

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

Offset: 0

Matthew A. Morgan, Oct 15 2017

This is the smallest ripple factor (a constant) for which the prototype elements of the seventh-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.

			0.2187077239...

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

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

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.

R:= RealField(); Sqrt(Exp(4*Argtanh(Exp(-2*7*Argsinh(Sqrt(1/2* Sin(Pi(R)/7)*Tan(Pi(R)/7))))))-1); // G. C. Greubel, Feb 15 2018

RealDigits[Sqrt[Exp[4 ArcTanh[Exp[-2*7*ArcSinh[Sqrt[1/2*Sin[Pi/7] Tan[Pi/7]]]]]] - 1], 10, 100][[1]]

sqrt(exp(4*atanh(exp(-2*7*asinh(sqrt(1/2*sin(Pi/7)*tan(Pi/7))))))-1) \\ Michel Marcus, Oct 16 2017

Equals sqrt(exp(4*arctanh(exp(-2*7*arcsinh(sqrt(1/2*sin(Pi/7)tan(Pi/7))))))-1).

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

Original entry on oeis.org

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

Offset: 0

Matthew A. Morgan, Oct 15 2017

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.

			0.2192047733...

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

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

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.

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

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

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

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

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

Original entry on oeis.org

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

Matthew A. Morgan, Oct 15 2017

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.

			0.2194869308...

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

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

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.

R:= RealField(); Sqrt(Exp(4*Argtanh(Exp(-Pi(R)*Sqrt(2))))-1); // G. C. Greubel, Feb 16 2018

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

sqrt(exp(4*atanh(exp(-Pi*sqrt(2))))-1) \\ Michel Marcus, Oct 15 2017

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

A293768 Continued fraction expansion of the minimum ripple factor for a fifth-order, reflectionless, Chebyshev filter.

Original entry on oeis.org

0, 4, 1, 1, 1, 1, 1, 3, 5, 1, 10, 5, 2, 2, 1, 3, 5, 4, 2, 1, 1, 3, 1, 3, 1, 8, 8, 164, 2, 2, 5, 4, 19, 1, 2, 74, 1, 1, 2, 1, 9, 1, 3, 1, 2, 2, 2, 3, 1, 1, 15, 1, 2, 1, 2, 3, 1, 45, 2, 4, 1, 1, 8, 1, 4, 2, 5, 1, 1, 2, 11, 1, 8, 1, 4, 4, 1, 1, 1, 1, 68, 10, 2, 4, 8, 1, 3, 5, 1, 25, 3, 1, 1, 8, 5, 81, 2, 1, 1, 2, 1, 868, 1, 4, 1

Offset: 0

Matthew A. Morgan, Oct 15 2017

This is the smallest ripple factor (a constant) for which the prototype elements of the fifth-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.

			1/(4 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + 1/(3 + 1/(5 + 1/(1+...

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

G. C. Greubel, Table of n, a(n) for n = 0..9999

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.

R:= RealField(); ContinuedFraction(Sqrt(Exp(4*Argtanh(Exp(-10* Argsinh(Sqrt(Sin(Pi(R)/5)*Tan(Pi(R)/5)/2))))) - 1)); // G. C. Greubel, Feb 16 2018

ContinuedFraction[Sqrt[Exp[4 ArcTanh[Exp[-2*5*ArcSinh[Sqrt[1/2*Sin[Pi/5] Tan[Pi/5]]]]]] - 1], 130]

contfrac( sqrt(exp(4*atanh(exp(-10*asinh(sqrt(sin(Pi/5)*tan(Pi/5)/2))))) - 1) ) \\ G. C. Greubel, Feb 16 2018

Offset changed by Andrew Howroyd, Aug 10 2024

A293769 Continued fraction expansion of the minimum ripple factor for a seventh-order, reflectionless, Chebyshev filter.

Original entry on oeis.org

0, 4, 1, 1, 2, 1, 22, 2, 1, 1, 1, 2, 81, 4, 1, 1, 2, 20, 1, 1, 1, 5, 2, 5, 3, 4, 1, 2, 1, 6, 2, 1, 15, 1, 2, 1, 2, 1, 1, 23, 1, 1, 1, 4, 1, 42, 1, 11, 1, 1, 1, 7, 1, 1, 5, 30, 1, 2, 7, 5, 2, 6, 1, 1, 1, 5, 5, 5, 7, 2, 1, 8, 6, 5, 1, 1, 2, 36, 34, 1, 3, 1, 1, 2, 1, 3, 2, 1, 1, 1, 5, 4, 47, 1, 3, 2, 1, 2, 2, 1, 1, 7, 1, 3, 1

Offset: 0

Matthew A. Morgan, Oct 15 2017

This is the smallest ripple factor (a constant) for which the prototype elements of the seventh-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.

			1/(4 + 1/(1 + 1/(1 + 1/(2 + 1/(1 + 1/(22 + 1/(2 + 1/(1 + 1/(1+...

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

G. C. Greubel, Table of n, a(n) for n = 0..9999

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.

R:= RealField(); ContinuedFraction(Sqrt(Exp(4*Argtanh(Exp(-14* Argsinh(Sqrt(Sin(Pi(R)/7)*Tan(Pi(R)/7)/2))))) - 1)); // G. C. Greubel, Feb 16 2018

ContinuedFraction[Sqrt[Exp[4 ArcTanh[Exp[-2*7*ArcSinh[Sqrt[1/2*Sin[Pi/7] Tan[Pi/7]]]]]] - 1], 130]

contfrac( sqrt(exp(4*atanh(exp(-14*asinh(sqrt(sin(Pi/7)*tan(Pi/7)/2))))) - 1) ) \\ G. C. Greubel, Feb 16 2018

Offset changed by Andrew Howroyd, Aug 10 2024

A293770 Continued fraction expansion of the minimum ripple factor for a ninth-order, reflectionless, Chebyshev filter.

Original entry on oeis.org

0, 4, 1, 1, 3, 1, 1, 6, 2, 7, 1, 1, 8, 3, 2, 5, 1, 2, 1, 13, 1, 2, 1, 10, 1, 1, 78, 7, 1, 11, 4, 2, 7, 4, 20, 1, 3, 3, 1, 18, 55, 1, 11, 2, 12, 1, 6, 1, 11, 1, 11, 1, 2, 1, 2, 2, 11, 3, 15, 1, 29, 2, 1, 1, 5, 1, 3, 1, 1, 1, 16, 1, 14, 1, 7, 1, 19, 2, 8, 2, 3, 14, 1, 4, 1, 28, 5, 11, 2, 1, 2, 255, 5, 1, 1, 1, 1, 5, 1, 3, 2, 2

Offset: 0

Matthew A. Morgan, Oct 15 2017

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.

			1/(4 + 1/(1 + 1/(1 + 1/(3 + 1/(1 + 1/(1 + 1/(16+ 1/(2 + 1/(7+...

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

G. C. Greubel, Table of n, a(n) for n = 0..9999

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.

R:= RealField(); ContinuedFraction(Sqrt(Exp(4*Argtanh(Exp(-18* Argsinh(Sqrt(Sin(Pi(R)/9)*Tan(Pi(R)/9)/2))))) - 1)); // G. C. Greubel, Feb 16 2018

ContinuedFraction[Sqrt[Exp[4 ArcTanh[Exp[-2*9*ArcSinh[Sqrt[1/2*Sin[Pi/9] Tan[Pi/9]]]]]] - 1], 130]

contfrac( sqrt(exp(4*atanh(exp(-18*asinh(sqrt(sin(Pi/9)*tan(Pi/9)/2))))) - 1) ) \\ G. C. Greubel, Feb 16 2018

Offset changed by Andrew Howroyd, Aug 10 2024

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

Original entry on oeis.org

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, 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, 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

Offset: 0

Matthew A. Morgan, Oct 18 2017

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.

			1/(4 + 1/(1 + 1/(1 + 1/(3 + 1/(1 + 1/(22 + 1/(1 + 1/(3 + 1/(3 +...

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

G. C. Greubel, Table of n, a(n) for n = 0..9999

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.

R:= RealField();ContinuedFraction(Sqrt(Exp( 4*Argtanh(Exp (-(Pi(R)*Sqrt(2))))) - 1)); // Michel Marcus, Feb 17 2018

ContinuedFraction[Sqrt[Exp[4 ArcTanh[Exp[-(Pi Sqrt[2])]]] - 1],130]

contfrac(sqrt(exp(4*atanh(exp(-Pi*sqrt(2)))) - 1)) \\ Michel Marcus, Feb 17 2018

Offset changed by Andrew Howroyd, Aug 10 2024

A344362 Decimal expansion of (5^(1/4) + 5^(-1/4))/2.

Original entry on oeis.org

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

Offset: 1

Daniel Carter, May 15 2021

Solution for x in the system {x = 1/y + 1/z, y = x + 1/z, z = y + 1/x}. The corresponding values of y and z are 5^(1/4) and (5^(1/4) + 5^(3/4))/2.

The smallest aspect ratio of a set of three rectangles which have the property that any two of them can be scaled, rotated, and joined at an edge to obtain a rectangle with the third aspect ratio. The other two aspect ratios are given in the comment above.

			1.082044543098821282957566033699780665875...

Cf. A011003, A344363, A293409, A176015.

RealDigits[Cosh[Log[5]/4], 10, 120][[1]] (* Amiram Eldar, Jun 29 2023 *)

solve(x=1, 2, 5*x^4 - 5*x^2 - 1) \\ Hugo Pfoertner, May 16 2021

my(c=50+30*quadgen(20)); a_vector(len) = digits(sqrtint(floor(c*100^(len-2)))); \\ Kevin Ryde, May 28 2021

Equals 5*A293409.
Equals sqrt(A176015).
Equals cosh(log(5)/4). - Vaclav Kotesovec, May 28 2021

cp's OEIS Frontend

A293415 Decimal expansion of the minimum ripple factor for a seventh-order, reflectionless, Chebyshev filter.

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

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

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A293768 Continued fraction expansion of the minimum ripple factor for a fifth-order, reflectionless, Chebyshev filter.

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A293769 Continued fraction expansion of the minimum ripple factor for a seventh-order, reflectionless, Chebyshev filter.

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A293770 Continued fraction expansion of the minimum ripple factor for a ninth-order, reflectionless, Chebyshev filter.

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