A040021 -id:A040021 - cp's OEIS frontend

A010482 Decimal expansion of square root of 27.

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

Offset: 1

N. J. A. Sloane

Continued fraction expansion is 5 followed by {5, 10} repeated (A040021). - Harry J. Smith, Jun 04 2009
6 + sqrt(27) represents the surface of a dodecahedron of side equal to one. S = 3*a^2(2 + sqrt(3)) with a = 1. - Vincenzo Librandi, Jul 10 2010
sqrt(27) is the perimeter of an equilateral triangle whose incircle's diameter is 1. - Martin Janecke, May 31 2016
If r = 2*a * sin(3t)/sin(2t) and x*(x^2+y^2) = a * (3x^2-y^2) are respectively a polar equation and a Cartesian equation of the Maclaurin trisectrix, then sqrt(27) * a^2 = area of the loop of this trisectrix = area between the curve and its asymptote (see Mathcurve link). - Bernard Schott, Jul 14 2020
Area of a regular hexagon with side length sqrt(2). - Christoph B. Kassir, Sep 29 2022
The solution of x^sqrt(3)=sqrt(3)^x, see e.g. A360148. - R. J. Mathar, Mar 24 2023
Surface area of a snub disphenoid (Johnson solid J_84) with unit edges. - Paolo Xausa, Aug 02 2025

			5.196152422706631880582339024517617100828415761431141884167420938355799....

Harry J. Smith, Table of n, a(n) for n = 1..20000
Robert Ferréol, MacLaurin trisectrix, Mathcurve.
Wikipedia, Snub disphenoid.
Index to sequences related to curves.
Index entries for algebraic numbers, degree 2

Cf. A040021 (continued fraction), A248254 (Egyptian fraction).

Cf. A104956 (half), A002194 (sqrt(3)).

RealDigits[N[Sqrt[27], 200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 22 2011 *)

default(realprecision, 20080); x=sqrt(27); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b010482.txt", n, " ", d));  \\ Harry J. Smith, Jun 04 2009

Equals 3*sqrt(3) = 3 * A002194. - Bernard Schott, Jul 14 2020

Equals 2 * A104956. - Christoph B. Kassir, Oct 02 2022

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

Original entry on oeis.org

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

A190567 Continued fraction expansion of 46*sqrt(46).

Original entry on oeis.org

311, 1, 76, 1, 622, 1, 76, 1, 622, 1, 76, 1, 622, 1, 76, 1, 622, 1, 76, 1, 622, 1, 76, 1, 622, 1, 76, 1, 622, 1, 76, 1, 622, 1, 76, 1, 622, 1, 76, 1, 622, 1, 76, 1, 622, 1, 76, 1, 622, 1, 76, 1, 622, 1, 76, 1, 622, 1, 76, 1, 622, 1, 76, 1, 622, 1, 76, 1, 622, 1, 76, 1, 622

Offset: 0

Bruno Berselli, May 13 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
G. Xiao, Contfrac.
Index entries for continued fractions for constants.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,1).

Cf. A010136; A040005, A040021, A010186, A040201, A040324, A040489, A040968.

[311] cat &cat[ [1,76,1,622]: n in [1..18] ];

I:=[311,1,76,1,622]; [n le 5 select I[n] else Self(n-4): n in [1..80]]; // Vincenzo Librandi, Jun 14 2013

ContinuedFraction[46 Sqrt[46], 80] (* or *) PadRight[{311}, 80, {622, 1, 76, 1}]
CoefficientList[Series[(311 + x + 76 x^2 + x^3 + 311 x^4) / (1 - x^4), {x, 0, 100}], x] (* Vincenzo Librandi, Jun 14 2013 *)

a(n)=if(n,[622,1,76,1][n%4+1],311) \\ Charles R Greathouse IV, May 13 2011

G.f.:  (311+x+76*x^2+x^3+311*x^4)/(1-x^4).
a(n) = 1+3*(1+(-1)^n)*(116+91*i^n)/2 with n>0, i=sqrt(-1) and a(0)=311.
a(n) = (-1513*(n mod 4)+575*((n+1) mod 4)+125*((n+2) mod 4)+2213*((n+3) mod 4))/12  for n>0.
a(n) = a(n-4), n>=5. - Vincenzo Librandi, Jun 14 2013

cp's OEIS Frontend

A010482 Decimal expansion of square root of 27.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs