A020784 -id:A020784 - cp's OEIS frontend

A293409 Decimal expansion of the minimum ripple factor for a fifth-order, reflectionless, Chebyshev filter.

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

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.

			0.216408908619764256591513206739956133175149494926718391028657695319669...

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
Index entries for algebraic numbers, degree 2.

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.

Sqrt(1/50 + 3/(50*Sqrt(5))); // G. C. Greubel, Feb 15 2018

RealDigits[Sqrt[1/50+3/(50*Sqrt[5])], 10, 100][[1]]

sqrt(1/50+3/(50*sqrt(5))) \\ Michel Marcus, Oct 16 2017

polrootsreal(3125*x^4-125*x^2-1)[2] \\ Charles R Greathouse IV, Feb 04 2025

Equals sqrt((3+sqrt(5))/(50*sqrt(5))).

Equals phi / 5^(5/4), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, May 28 2021

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

Original entry on oeis.org

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

A118273 Decimal expansion of (4/3)^(3/2).

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Apr 21 2006

The volume of the cube inscribed in the unit-radius sphere. - Amiram Eldar, Jun 02 2023

			1.539600717839002038...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 5.24, p. 412.

Elliott H. Lieb, Residual entropy of square ice, Phys. Rev. 162 (1967) 162-172.
James Propp, The many faces of alternating-sign matrices, 2002.
Scenta server, Lieb's square ice constant. [Broken link]
Eric Weisstein's World of Mathematics, Lieb's Square Ice Constant.
Wikipedia, Lieb's square ice constant.
Index entries for algebraic numbers, degree 2.

Cf. A020784, A054759.

Cf. A122553 (octahedron), A339259 (regular icosahedron), A363437 (regular tetrahedron), A363438 (regular dodecahedron).

evalf(sqrt((4/3)^3)) ; # R. J. Mathar, Mar 29 2012

RealDigits[(4/3)^(3/2),10,120][[1]] (* Harvey P. Dale, Aug 06 2015 *)

(4/3)^(3/2) \\ Stefano Spezia, Dec 15 2024

Equals 8 * A020784.

A165922 Decimal expansion of 2sqrt(3)/(9Pi).

Original entry on oeis.org

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

Offset: 0

Rick L. Shepherd, Sep 30 2009

The ratio of the volume of a regular tetrahedron to the volume of the circumscribed sphere. (The MathWorld link shows that the circumradius for a tetrahedron with side length a is a*sqrt(6)/4.)

			0.122517532315953788780294777402882098...

Frank Jackson and Eric W. Weisstein, Tetrahedron
Index entries for transcendental numbers

Cf. A000796, A002194, A010469, A049541, A020784.

RealDigits[(2Sqrt[3])/(9Pi),10,120][[1]] (* Harvey P. Dale, Nov 17 2013 *)

PARI
```
2*3^(-3/2)/Pi
```

2*sqrt(3)/(9*Pi) = A010469/(9*A000796) = (2/9)*A002194/A000796 = (2/9)*A002194*A049541 = 2*A020784/A000796 = 2*3^(-3/2)/Pi.

cp's OEIS Frontend

A293409 Decimal expansion of the minimum ripple factor for a fifth-order, reflectionless, Chebyshev filter.

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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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A165922 Decimal expansion of 2sqrt(3)/(9Pi).